ENGINEERING  LIBRARY 


A  COMPENDIUM  OF  SPHERICAL  ASTRONOMY 


A    COMPENDIUM 

OF 

SPHERICAL   ASTRONOMY 


WITH  ITS  APPLICATIONS  TO  THE  DETERMINATION 

AND   REDUCTION  OF  POSITIONS  OF 

THE  FIXED   STARS 


BY 

SIMON    NEWCOMB 


THE    MACMILLAN    COMPANY 

LONDON  :   MACMILLAN  AND  CO.,  LTD. 
1906 

All  rights  reserved 


ENGINEERING  LIBRARY 


•      ••.»«••.•»»•••         •      • 

•*„*•  •  •/•*»••    *  •  •  T      •  •*'/.»    • 
•    •••    ••"•*•      •«».•«*•   ,• 


GLASGOW:   PRINTED  AT  THE  UNIVERSITY  PRESS 

BY    ROBERT   MACLEHOSE   AND   CO.    LTD. 


PREFACE. 

THE  present  volume  is  the  first  of  a  projected  series  having 
the  double  purpose  of  developing  the  elements  of  Practical  and 
Theoretical  Astronomy  for  the  special  student  of  the  subject, 
and  of  serving  as  a  handbook  of  convenient  reference  for 
the  use  of  the  working  astronomer  in  applying  methods  and 
formulae.  The  plan  of  the  series  has  been  suggested  by  the 
author's  experience  as  a  teacher  at  the  Johns  Hopkins  Uni- 
versity, and  as  an  investigator.  The  first  has  led  him  to  the 
view  that  the  wants  of  the  student  are  best  subserved  by  a 
quite  elementary  and  condensed  treatment  of  the  subject, 
without  any  attempt  to  go  far  into  details  not  admitting  of 
immediate  practical  application.  As  an  investigator  he  has 
frequently  been  impressed  with  the  amount  of  time  consumed 
in  searching  for  the  formulae  and  data,  even  of  an  elementary 
kind,  which  should  be,  in  each  case,  best  adapted  to  the  work 
in  hand. 

The  most  urgent  want  which  the  work  is  intended  to 
supply  is  that  of  improved  methods  of  deriving  and  reducing 
the  positions,  and  proper  motions  of  the  fixed  stars.  Modifica- 
tions of  the  older  methods  are  made  necessary  by  the  long 
period,  150  years,  through  which  positions  of  the  stars  now 
have  to  be  reduced,  and  by  the  extension  of  astrometrical 
and  statistical  researches  to  a  great  and  constantly  increasing 
number  of  telescopic  stars.  Especial  attention  has  therefore 


vi  PEEFACE 

been  given  to  devising  the  most  expeditious  and  rigorous 
methods  of  trigonometric  reduction  of  star  positions,  and  to 
the  construction  of  tables  to  facilitate  the  work. 

Other  features  of  the  work  are :  A  condensed  treatment  of 
the  theory  of  errors  of  observation  and  of  the  method  of 
least  squares ;  an  attempt  to  present  the  theory  of  astronomical 
refraction  in  a  concise  and  elementary  form  without  detract- 
ing from  rigour  of  treatment;  a  new  development  of  the 
theory  of  precession,  now  rendered  necessary  by  the  long 
period  through  which  star  places  have  to  be  reduced ;  the 
basing  of  formulae  relating  to  celestial  coordinates  on  the 
new  values  of  the  constants  now  used  in  the  national 
ephemerides;  a  concise  development  of  the  rigorous  theory  of 
proper  motions ;  the  trigonometric  reduction  of  polar  stars 
to  apparent  place,  and  the  development  of  what  the  author 
deems  the  most  advantageous  methods  of  correcting  and  com- 
bining observed  positions  of  stars  as  found  in  catalogues. 

Although  the  theory  of  astronomical  instruments  is  not  in- 
cluded within  the  scope  of  the  present  work,  it  is  necessary, 
in  using  star  catalogues,  to  understand  the  methods  of  deriving 
the  results  therein  found  from  observations.  The  principles 
of  the  ideal  transit  instrument  and  meridian  circle,  omitting 
all  details  arising  from  imperfections  of  the  instrument,  are 
elegant  and  simple,  and  at  the  same  time  sufficient  for  the 
purpose  in  question.  They  are  therefore  briefly  set  forth  in 
the  chapter  on  deriving  mean  positions  of  stars  from  meridian 
observations. 

A  pedagogical  feature  of  the  work  is  the  effort  to  give 
objective  reality  to  geometric  conceptions  in  every  branch  of 
the  subject.  The  deduction  of  results  by  purely  algebraic 
processes  is  therefore  always  supplemented,  when  convenient, 
by  geometric  construction.  Whenever  such  a  construction  is 


PREFACE  vii 

represented  on  the  celestial  sphere,  the  latter  is,  in  the  absence 
of  any  reason  to  the  contrary,  shown  as  seen  from  the 
centre,  so  that  the  figure  shows  the  sky  as  one  actually 
looks  up  at  it.  Exceptions  to  this  are  some  times  necessary 
when  planes  and  axes  of  reference  have  to  be  studied  in 
connection  with  their  relation  to  the  sphere. 

A  similar  feature,  which  may  appear  subject  to  criticism, 
is  the  subordination  of  logical  order  of  presentation  to 
the  practical  requirements  of  the  student  mind.  While  the 
method  of  first  developing  a  subject  in  its  general  form 
and  then  branching  out  into  particulars  has  been  fol- 
lowed whenever  it  seemed  best  so  to  do,  there  are  many 
cases  in  which  special  forms  of  a  theory  are  treated  in 
advance  of  the  general  form,  the  object  being  to  prepare  the 
mind  of  the  student  for  the  more  ready  apprehension  of  the 
general  theory. 

On  the  other  hand,  in  order  to  lessen  discontinuity  of 
treatment,  the  policy  has  been  adopted  of  relegating  to  an 
Appendix  all  the  tables  and  many  of  the  formulae  of  which 
most  use  is  made.  The  choice  of  subjects  for  the  Appendix 
is  made  from  a  purely  practical  point  of  view,  the  purpose 
being  to  include  those  tables,  formulae,  and  data  of  most 
frequent  application. 

The  "  Notes  and  References "  at  the  end  of  most  of  the 
chapters  do  not  aim  at  logical  or  practical  completeness. 
They  embody  such  matters  of  interest,  historical  or  otherwise, 
and  such  citations  of  literature,  as  the  author  hopes  may 
be  most  useful  to  the  student  or  the  working  astronomer. 
The  list  of  Star  Catalogues  of  precision  at  the  end  of  the 
last  chapter  is,  however,  intended  to  be  as  complete  as  it 
was  found  practicable  to  make  it ;  but  even  here  it  may  well 
be  that  important  catalogues  have  been  overlooked. 


viii  PKEFACE 

The  habit  on  the  part  of  computers  of  using  logarithmic 
tables  to  more  decimals  than  are  necessary  is  so  common 
that  tables  to  three  decimals  only  are  not  always  at  hand. 
The  Appendix  therefore  concludes  with  three-place  tables  of 
logarithms  and  trigonometric  functions.  These  will  suffice 
for  the  ordinary  reduction  of  stars  to  apparent  place,  and  many 
similar  computations  which  have  to  be  executed  on  a  large 
scale. 

WASHINGTON,  March,  1906. 


CONTENTS. 

PART  I. 
PRELIMINARY  SUBJECTS. 

CHAPTER  I. 

PAGE 

INTRODUCTORY,  -  3. 

1:  Use  of  finite  quantities  as  infinitesimals.  2.  Use  of  small  angles 
for  their  sines  or  tangents.  3,  4.  Unavoidable  errors  in  computation. 
5.  Derivatives,  speeds,  and  units.  6.  Differential  relations  between 
the  parts  of  a  spherical  triangle.  7.  Differential  spherical  trigono- 
metry. 

NOTES  AND  REFERENCES,  13. 

CHAPTEE  II. 

DIFFERENCES,  INTERPOLATION,  AND  DEVELOPMENT,  -  15 

8,  9.  Differences  of  various  orders.  10.  Detecting  errors  by  dif- 
ferencing. 11.  Use  of  higher  orders  of  differences  in  interpolation. 
12.  Transformations  of  the  formula  of  interpolation.  13.  Stirling's 
formula  of  interpolation.  14.  Bessel's  formula  of  interpolation. 
15.  Interpolation  to  halves.  16.  Interpolation  to  thirds.  17.  Inter- 
polation to  fourths.  18.  Interpolation  to  fifths.  19.  Numerical 
differentiation  and  integration.  20.  Development  in  powers  of  the 
time. 

NOTES  AND  REFERENCES,  -  3& 


jc  CONTENTS 

CHAPTER   III. 

PAGE 

THE  METHOD  OF  LEAST  SQUARES,  40 

SECTION  I.     Mean  Values  of  Quantities. 

21,  22.  Distinction  of  systematic  and  fortuitous  errors.  23.  The 
arithmetical  mean  and  the  sum  of  the  squares  of  residuals.  24.  The 
probable  error.  25.  Weighted  means.  26.  Relation  of  probable 
errors  to  weights.  27.  Modification  of  the  principle  of  least  squares 
when  the  weights  are  different.  28.  Adjustment  of  quantities. 

SECTION  II.     Determination  of  Probable  Errors. 

29.  Of  probable  and  mean  errors.  30.  Statistical  distribution  of 
errors  in  magnitude.  31.  Method  of  determining  mean  or  probable 
errors.  32.  Case  of  unequal  weights.  33.  To  find  the  probable 
mean  error  when  the  weights  are  unequal. 

SECTION  III.     Equations  of  Condition. 

34.  Elements  and  variables.  35.  Method  of  correcting  provisional 
elements.  36.  Conditional  and  normal  equations.  37.  Solution  of 
the  normal  equations.  38.  Weights  of  unknown  quantities  whose 
values  are  derived  from  equations  of  condition.  39.  Special  case  of 
a  quantity  varying  uniformly  with  the  time.  40.  The  mean  epoch. 

NOTES  AND  REFERENCES,  •  •    •        84 


PART  II. 

THE  FUNDAMENTAL  PRINCIPLES  OF  SPHERICAL 
ASTRONOMY. 

CHAPTER  IV. 
SPHERICAL  COORDINATES,  -        -      87 

SECTION  I.     General  Theory. 

42,  43,  44.  The  celestial  sphere.  45.  Special  fundamental  planes 
and  their  associated  concepts.  46.  Special  systems  of  coordinates. 
47.  Relations  of  spherical  and  rectangular  coordinates.  48.  Differ- 
entials of  rectangular  and  spherical  coordinates.  49.  Relations  of 
equatorial  and  ecliptic  coordinates. 


CONTENTS  xi 

SECTION  II.     Problems  and  Applications  of  the  Theory  of 

Spherical  Coordinates.  PAGE 

51.  To  convert  longitude  and  latitude  into  R.A.  and  Dec.  52.  Use 
of  the  Gaussian  equations  for  the  conversion.  53.  Check  com- 
putations. 54.  Effect  of  small  changes  in  the  coordinates.  55. 
Geometric  construction  of  small  changes.  56,  57.  Position  angle 
and  distance. 

CHAPTER  V. 
THE  MEASURE  OF  TIME  AND  RELATED  PROBLEMS,  -  114 

SECTION  I.     Solar  and  Sidereal  Time. 

58.  Solar  and  sidereal  time.  59.  Relations  of  the  sidereal  and  solar 
day.  60.  Astronomical  mean  time.  61.  Time,  longitude,  and  hour 
angle.  62.  Absolute  and  local  time.  63.  Recapitulation  and  illus- 
tration. 64.  Effect  of  nutation.  65.  The  year  and  the  conversion 
of  mean  into  sidereal  time,  and  vice  versa, 

SECTION  II.     The  General  Measure  of  Time. 

66.  Time  as  a  flowing  quantity.  67.  Units  of  time  :  the  day  and 
year.  68.  The  solar  or  Besselian  year.  69.  Sidereal  time  of 
mean  noon. 

SECTION  III.     Problems  Involving  Time. 

70.  Problems  of  the  conversion  of  time.     71.   Related  problems. 

CHAPTER  VI. 
PARALLAX  AND  RELATED  SUBJECTS,  -        -    141 

SECTION  I.     Figure  and  Dimensions  of  the  Earth. 
72.    The  geoid.     73.    Local  deviation  of  the  plumb-line.     74.    Geo- 
centric and  astronomical  latitude      75.   Geocentric  coordinates  of  a 
station  on  the  earth's  surface.     76.  Dimensions  and  compression  of 
the  geoid. 

SECTION  II.     Parallax  and  Semi-diameter. 

78.  Parallax  in  altitude.  79.  Parallax  in  right  ascension  and 
declination.  80.  Transformed  expression  for  the  parallax.  81.  Mean 
parallax  of  the  moon.  82.  Parallaxes  of  the  sun  and  planets. 
83.  Semi-diameters  of  the  moon  and  planets. 


xii  CONTENTS 

CHAPTER   VII. 

PAGE 

ABERRATION,  .    160 

84.  Law  of  aberration.  85.  Reduction  to  spherical  coordinates. 
86.  The  constant  of  aberration  and  related  constants.  87.  Aberra- 
tion in  right  ascension  and  declination.  88.  Diurnal  aberration. 
89.  Aberration  when  the  body  observed  is  itself  in  motion.  90.  Case 
of  rectilinear  and  uniform  motion.  91.  Aberration  of  the  planets. 

CHAPTER   VIII. 
ASTRONOMICAL  REFRACTION,  173 

SECTION  I.     The  Atmosphere  as  a  Refracting  Medium. 

92.  Astronomical  refraction  in  general.  93.  Density  of  the 
atmosphere  as  a  function  of  the  height.  95,  96.  Numerical  data  and 
results.  97.  General  view  of  requirements.  98.  Density  at  great 
heights.  99.  Hypothetical  laws  of  atmospheric  density.  100.  De- 
velopment of  the  hypotheses.  101.  Comparison  of  densities  of  the 
air  at  different  heights  on  the  two  best  hypotheses. 

SECTION  II.     Elementary  Exposition  of  Atmospheric  Refraction. 

102.  General  view.  103.  Refraction  at  small  zenith  distances. 
104.  Differential  of  the  refraction.  105.  Relation  of  density  to 
refractive  index.  106.  Form  in  which  the  refraction  is  expressed. 
107.  Practical  determination  of  the  refraction.  108.  Curvature  of  a 
refracted  ray.  109.  Distance  and  dip  of  the  sea  horizon. 

SECTION  III.     General  Investigation  of  Astronomical  Refraction* 

110.  Fundamental  equation  of  refraction.  111.  Transformation  of 
the  differential  equation.  112.  The  integration.  113,114.  Develop- 
ment of  the  refraction.  115.  Development  on  Newton's  hypothesis. 
116.  Development  on  Ivory's  hypothesis.  117.  Construction  of 
tables  of  refraction.  118.  Development  of  factors. 

NOTES  AND  REFERENCES  TO  REFRACTION,     -  223 


CHAPTER  IX. 
PRECESSION  AND  NUTATION,      ...  -    225 


CONTENTS  xiii 

SECTION  I.     Laws  of  the  Precessional  Motion. 

PAGE 

119.  Fundamental  definitions.  120.  Fundamental  conceptions. 
121.  Motion  of  the  celestial  pole.  122.  Motion  of  the  ecliptic. 
123.  Numerical  computation  of  the  motion  of  the  ecliptic.  124.  Com- 
bination of  the  processional  motions.  125.  Expressions  for  the 
instantaneous  rates  of  motion.  126.  Numerical  values  of  the  pre- 
cessional  motions  and  of  the  obliquity. 

SECTION  II.     Relative  Positions  of  the  Equator  and  Equinox  at  Widely 
Separated  Epochs. 

127.  Definitions  of  angles.  128.  Numerical  approximations  to  the 
position  of  the  pole.  129.  Numerical  value  of  the  planetary  pre- 
cession. 130.  Auxiliary  angles.  131.  Computation  of  angle  between 
the  equators. 

SECTION  III.     Nutation. 

132.  Motion  of  nutation.  133.  Theoretical  relations  of  precession 
and  nutation. 

NOTES  AND  REFERENCES  TO  PRECESSION  AND  NUTATION,          -      253 


PART  III. 

REDUCTION   AND   DETERMINATION   OF  POSITIONS 
OF   THE  FIXED  STARS. 

CHAPTER  X. 

REDUCTION  OF  MEAN  PLACES  OF  THE  FIXED  STARS  FROM  ONE  EPOCH 
TO  ANOTHER,  259 

135.  System  of  reduction  explained. 

SECTION  I.     The  Proper  Motion  of  the  Stars. 

136.  Law  of  proper  motion.     137.  Reduction  for  proper  motion. 

SECTION  II.     Trigonometric  Reduction  for  Precession. 

138.  Rigorous  formulae  of  reduction.     139.    Geometric  signification 
of  the  constants.      140.   Approximate  formulae.      141.   Construction 


v  CONTENTS 

PAGE 

of  tables  for  the  reduction.  142.  Reduction  of  the  declination. 
143.  Failure  of  the  approximation  near  the  pole.  144.  Reduction  of 
the  proper  motion. 

SECTION  III.     Development  of  the  Coordinates  in  Powers  of  the  Time. 

145.  The  annual  rates  of  motion.  146.  The  secular  variations. 
147.  Use  of  the  century  as  the  unit  of  time.  148.  The  third  term  of 
the  reduction.  149.  Precession  in  longitude  and  latitude. 

NOTES  AND  REFERENCES,  -  -  -          -          -          -     288 


CHAPTER   XL 
REDUCTION  TO  APPARENT  PLACE,     -_  •>    289 

SECTION  I.     Reduction  to  Terms  of  the  First  Order. 

150.  Reduction  for  nutation.  151.  Nutation  in  R.A.  and  Dec. 
152.  Reduction  for  aberration.  153.  Reduction  for  parallax. 
154.  Combination  of  the  reductions.  155.  Independent  day  numbers. 

SECTION  II.     Rigorous  Reduction  for  Close  Polar  Stars. 

156.  Cases  when  a  rigorous  reduction  is  necessary.  157.  Trigono- 
metric reduction  for  nutation.  158.  Trigonometric  reduction  for 
aberration. 

SECTION  III.     Practical  Methods  of  Reduction. 

159.  Three  classes  of  terms.  160.  Treatment  of  the  small  terms  of 
nutation.  161.  Development  of  the  reduction  to  terms  of  the  second 
order.  162.  Precession  and  nutation.  163.  Aberration.  164.  Effect 
of  terms  of  the  second  order  near  the  pole. 

SECTION  IV.     Construction  of  Tables  of  the  Apparent  Places  of 

Fundamental  Stars. 

165.  Fundamental  stars  defined.  166.  Construction  of  tables  of 
apparent  places.  167.  Adaptation  of  the  tables  to  any  meridian. 

NOTES  AND  REFERENCES,-  -     315 


CHAPTER   XII. 

METHOD  OF  DETERMINING  THE  POSITIONS  OF  STARS  BY  MERIDIAN 
OBSERVATIONS,  317 

168.  Differential  and  fundamental  determinations.     169.   The  ideal 
transit  instrument  and  clock. 


CONTENTS 

SECTION  I.     Method  of  Determining  Right  Ascensions. 

170.  Principles  of  the  ideal  method.  171.  Practical  method  of 
determining  right  ascensions.  172.  Elimination  of  systematic  errors. 
173.  Reference  to  the  sun — the  equinoxial  error.  174.  Question  of 
policy.  175.  The  Greenwich  method. 

SECTION  II.     The  Determination  of  Declinations. 

176.  The  ideal  meridian  circle.  177.  Principles  of  measurement. 
178.  Differential  determinations  of  declination.  179.  Systematic 
errors  of  the  method. 


CHAPTER  XIII. 

METHODS  OF  DERIVING  THE  POSITIONS  AND  PROPER  MOTIONS  OF 
THE  STARS  FROM  PUBLISHED  RESULTS  OF  OBSERVATIONS,       339 

SECTION  I.     Historical  Review. 

180.  The  Greenwich  Observations.  181.  The  German  School. 
182.  The  Poulkova  Observatory.  183.  Observatories  of  the  southern 
hemisphere.  184.  Miscellaneous  observations.  185.  Observations 
of  miscellaneous  stars. 

SECTION  II.     Reduction  of  Catalogue  Positions  of  Stars  to  a  Homogeneous 

System. 

186.  Systematic  differences  between  catalogues.  187.  Systematic 
corrections  to  catalogue  positions.  188.  Form  of  the  systematic 
corrections.  189.  Method  of  finding  corrections.  190.  Distinction 
of  systematic  from  fortuitousdifferenc.es.  191.  Existing  fundamental 
systems. 

SECTION  III.     Methods  of  Combining  Star  Catalogues. 

192.  Use  of  star  catalogues.  193.  Preliminary  reductions.  194.  The 
two  methods  of  combination.  195.  Development  of  first  method. 
196.  Formation  and  solution  of  the  equations.  197.  Use  of  the  central 
date.  198.  Method  of  correcting  provisional  data.  199.  Special 
method  for  close  polar  stars. 

NOTES  AND  REFERENCES,     -  -378 

LIST  OF  INDEPENDENT  STAR  CATALOGUES,     -  380- 

CATALOGUES  MADE  AT  NORTHERN  OBSERVATORIES,  -     380 

CATALOGUES  MADE  AT  TROPICAL  AND  SOUTHERN  OBSERVATORIES.  -      386 


xvi  CONTENTS 


APPENDIX. 

PAGE 

EXPLANATIONS  or  THE  TABLES  OF  THE  APPENDIX,  389 

I.     CONSTANTS  AND  FORMULAE  IN  FREQUENT  USE,-  393 

A.  Constants.  B.  Formulae  for  the  solution  of  Spherical 
Triangles.  C.  Differentials  of  the  parts  of  a  Spherical 
Triangle. 

II.     TABLES  RELATING  TO  TIME  AND  ARGUMENTS  FOR  STAR  REDUCTIONS,    397 
Table  I.    Days  of  the  Julian  Period. 

II.    Conversion  of  Mean  and  Sidereal  Time. 

III.  Time  into  Arc  and  vice  versa. 

IV.  Decimals  of  Day  to  Hours,  Minutes,  etc. 
V.-VII.    The  Solar  Year,  Lunar  Arguments. 

III.     CENTENNIAL  RATES  OF  THE  PRECESSIONAL  MOTIONS,  -  -     406 

Table  VIII.    Centennial  Precessions,  1750-1900.      Formulae  for 

Precession  in  R.A.  and  Dec. 
IX. -X.    Secular  Variations  of  Precessions. 

IV.     TABLES  AND  FORMULAE  FOR  THE  TRIGONOMETRIC  REDUCTION  OF 

MEAN  PLACES  OF  STARS,      -  412 

General  Expressions  for  the  Constants  of  Reduction. 
Table  XL   Special  of  Constants  for  1875  and  1900.     Precepts 

for  the  Trigonometric  Reduction. 
Tables  XII. -XVII.  Tables  for  the  Reduction. 

V.     REDUCTION  OF  THE  STRUVE-PETERS  PRECESSIONS  TO  THOSE   NOW 

ADOPTED,  428 

Tables  XVIII. -XIX.  Tables  for  the  Reduction. 

VI.     CONVERSION  OF  LONGITUDE  AND  LATITUDE  INTO  R.A.  AND  DEC.,      429 
Table  XX.   Tables  for  the  Conversion. 

XXI.    Conversion  of  Small  Changes. 

VII.     Table  XXII.     APPROXIMATE  REFRACTIONS,  433 

VIII.     COEFFICIENTS  OF  SOLAR  AND  LDNAR  NUTATION,  434 

IX.     Tables  XXIII. -XXVI.    THREE-PLACE  LOGARITHMIC  AND  TRIGONO- 
METRIC TABLES,      ------  -     435 


INDEX  TO  THE  NOTATION. 


=  ,  the  symbol  of  identity,  signifying  that  the  symbol  following  it  is 
defined  by  words  or  expression  preceding  it.  It  may  commonly  be 
read  "which  let  us  call." 

D,,  a  derivative  as  to  the  time,  expressing  the  rate  of  increase  of  the 
quantity  following  it. 

0,  sun's  true  longitude. 

In  the  following  list  of  symbols  only  those  significations  are  given,  which 
are  extensively  used  in  the  work.  Those  used  only  for  a  temporary  or 
special  purpose  are  omitted. 

Roman-Italic  alphabet. 

a,  semi-major  axis  of  an  ecliptic  orbit;   the  equatorial  radius  of  the 

earth  ;  also,  reduced  E.A.,  defined  on  p.  266. 

b,  polar   radius   of    the   earth ;    barometric   pressure ;    latitude    of    a 

heavenly  body. 

c,  earth's  compression. 

a,  6,  c,  d  are  used  to  denote  the  Besselian  star-constants.     Chap.  XI. 

e,  probable  error  ;  eccentricity. 

/,  ratio  of  apparent  to  geocentric  distance. 

<7,  intensity  of  gravity. 

A,  seconds  of  time  in  unit  radius  ;  also,  west  hour-angle. 

£,  angle  between  two  positions  of  the  plane  of  the  ecliptic,  or  of  the 

pole  of  the  ecliptic. 

£,  the  rate  of  general  precession,  annual  or  centennial, 
m,  the  factor  of  tan  z  in  the  expression  for  the  refraction  ;   also,  the 

constant  part  of  the  reduction  of  the  R.A.  of  a  star  for  precession, 
m,  the  annual  rate  of  precession  in  Right  Ascension  ;  rac,  the  centennial 

rate  =  100m. 
n,  the    annual    rate    of    motion    of    the   celestial  pole:   ?ic=100n,  the 

centennial  motion. 


xviii  INDEX  TO  THE   NOTATION 

N,  NQ,  supplement  of  the  longitude  of  the  instantaneous  axis  of  rotation  of 
the   moving   ecliptic ;    also,   the    angle   which   the  direction   of 
proper  motion  makes  with  the  hour-circle  of  a  star. 
Nlt  supplement  of  the  longitude  of  the  node  of  the  ecliptic. 

jo,  speed  of  luni-solar  precession  on  the  fixed  ecliptic  of  the  date  ;  also, 
a  quantity  used  in  star-reductions  (p.  267). 

p,  the  absolute  constant  of  precession. 

r,  radius  vector. 

s,  angular  semidiameter  of  a  planet ;  also,  angular  distance. 

£,  time  expressed  in  years  or  shorter  units  ;  also,  mean  time. 

T,  time  expressed  in  terms  of  a  century  as  the  unit. 

v,  linear  velocity,  especially  of  a  star,  or  of  the  earth  in  its  orbit ;  also, 
angle  of  the  vertical. 

F,  velocity  of  light. 

w,  weight  of  an  observation  or  result. 

2,  zenith  distance. 

Greek  alphabet. 
OC,  Right  Ascension. 
ft,  latitude,  referred  to  the  ecliptic. 
OC,  ft,  y,  angles  made  by  a  line  with  rectangular  axes. 
8,  Declination  ;  symbol  for  increment  or  correction. 
A,  symbol  of  increment,   of   error,   or    of   correction ;   distance  of    a 

planet  from  the  earth. 
€,  obliquity  of  the  ecliptic  ;  mean  error. 
{,  £0,  angles  defining  the   relative  positions   of  the   mean   equator  and 

equinox  at  two  epochs.     §§127-130. 
#,  angle  between  two  positions  of  the  mean  equator. 
K,  constant  of  aberration  ;  also,  speed  of  angular  motion  of  the  ecliptic. 
A,  ecliptic  longitude  ;  terrestrial  longitude  ;  also,  planetary  precession. 
/A,  proper  motion  of  a  star  ;  also,  index  of  refraction  of  air. 
ju,a,         „  „         in  Right  Ascension. 

ju,fi,         „  „         in  Declination. 

TT,  parallax  ;  ratio  of  circumference  to  diameter. 
p,  distance  from  centre  of  earth  ;  radius  of  earth. 
T,  temperature  above  absolute  zero  ;  also,  sidereal  time. 
^,  total  luni-solar  precession  on  an  initial  fixed  ecliptic. 
<£,  astronomical  latitude  of  a  point  on  the  earth's  surface. 
<£',  geocentric  latitude  „  „  „ 

il,  longitude  of  the  moon's  node. 


PART  I. 
PRELIMINARY   SUBJECTS. 


N.S.A. 


CHAPTER  I. 
INTRODUCTORY. 

THIS  opening  chapter  is  devoted  to  certain  preliminary  matters 
which  can  better  find  a  place  here  than  elsewhere.  The  beginner 
in  astronomical  work  may  be  accustomed  only  to  those  modes 
of  mathematical  thought  and  investigation  which  are  formally 
rigorous.  He  has  now  to  enter  a  field  in  which,  owing  to  the 
concrete  form  of  the  subject-matter,  he  must  frequently  be  satis- 
fied with  approximations  to  a  rigorous  result,  and  a  consequent 
abatement  of  the  strictness  of  mathematical  demonstration.  One 
must  learn  to  work  in  this  field  without  sacrificing  rigor  of 
thought,  or  losing  sight  of  the  possible  deviations  of  the  results 
from  the  ideal  truth.  To  do  this,  we  give  examples  of  the  most 
common  cases  of  deviation  from  formal  precision  in  astronomical 
practice. 

1.   Use  of  finite  quantities  as  infinitesimals. 

The  omission  of  all  powers  of  a  small  quantity  above  the  first 
is  very  common  in  the  mathematical  methods  of  astronomy.  In 
this  case  we  are  said  to  treat  the  quantity  as  an  infinitesimal. 
The  practice  rests  on  the  following  basis  : 

Let  u  be  a  function  of  x, 


and  let  us  assign  to  x  an  increment  A#,  and  call  Au  the  corre- 
sponding increment  of  u.     The  new  value  of  u  will  be 

(1) 


INTRODUCTORY  [§  1. 


Developing  by  Taylor's  theorem,  we  have 

du  .        1  d2u 
Au  =  -j—  Ax  +  o 
cfo          2 


If  A#  is  below  a  certain  limit  of  magnitude,  and  the  differential 
coefficients  -r-  etc.  not  too  great,  the  second  and  following  terms 

of  this  development  may  be  omitted.     For  example,  let  Ax  be 
50"  =  0*000  24  in  arc.     Reduced  to  seconds  the  square  will  be 

Aa;2  =  0"-012. 

In  much  astronomical  work  an  error  of  0"'01  is  quite  unim- 
portant ;  indeed  cases  are  frequent  in  which  we  need  not  consider 
a  correction  so  small  as  0"'l  or  even  an  entire  second.  We  may 
extend  and  generalize  this  conclusion  as  follows  : 

If  the  second  derivative  does  not  exceed  unity,  we  may  use  the 
equation 

du  .  /nN 

Aw=^A;c'  ..............................  <2> 

dropping  the  higher  terms  of  the  series 

whenever  Aa?<   50"  if  an  error  of  +0"'01  is  unimportant, 
„  <150"        „        „        ±0*1       „ 

„  <5oo"      „      „      +r-o     „ 

If  the  second  derivative  exceeds  unity,  the  limits  must  be  reduced 
in  a  like  proportion. 

2.   Use  of  small  angles  for  their  sines  or  tangents. 

The  general  rule  embodied  in  (2)  leads  to  the  constant  use  of 
small  angles  themselves  instead  of  their  sines  or  tangents,  and 
to  putting  their  cosines  equal  to  1.  We  have,  by  well-known 
developments, 

sin  s  =  s  — 


Hence,  to  terms  of  the  third  order  in  s,  we  may  use  the  forms 

sin2s) 


§  2.]  USE   OF  SMALL  ANGLES  5 

These  equations  presuppose  that  the  angle  is  expressed  in  circular 
measure,  the  radian  being  the  unit.  But,  in  actual  computation, 
the  unit  is  nearly  always  the  degree,  minute,  or  second,  the  last 
being  the  usual  unit  for  small  angles.  The  fact  that  s  is 
expressed  in  seconds  may  be  indicated  by  two  accents.  When 
so  expressed,  it  may  be  reduced  to  circular  measure  by  multipli- 
cation by  the  angle  of  I"  expressed  in  that  measure,  which  is 
practically  the  same  as  sin  1".  Thus  we  have 

s  =  s"sinl". 

There  being  206  265"  (more  exactly  206  264"'806)  in  the  radian, 
we  have  s"  =  206  265"s  =  [5'314425]s, 

the  number  in  brackets  being  the  logarithm  of  the  factor.  This 
form  of  expressing  multiplication  by  a  factor  whose  logarithm 
only  is  given  is  very  common.  So,  putting  R  for  the  number 
206  264-806,  we  may  write  instead  of  (4) 

s"  =  R"  sin  s(l  +  J  sin2s)  j 
s''==E''tans(l--itanV)J' 

If  we  have  a  series  containing  various  powers  of  a  small  angle, 
the  practically  easiest  method  of  manipulating  it  is  to  reduce 
one  factor  of  each  term  to  seconds,  and  retain  the  others  in  the 
general  form.  For  example,  the  general  form 


becomes,  in  seconds,       s"  =  a"  +  bs"s  +  cs"s2  .  .  .  , 

where  s"  =  R"s,  a"  =  R"a,  while  s  and  s2  are  expressed  in  radians. 
Below  a  certain  limit  we  may  drop  the  factor  sin2s  from  the 
equations  (.3)  and  (4).    This  limit  is  that  below  which  the  product 


sxsns   or 

is  too  small  to  affect  our  result.     If  an  error  of  0"'01  is  unim- 
portant, the  upper  limit  for  s"  falls  below  that  value  for  which 


,     . 
which  gives  snrs  =  — 


6  INTRODUCTOKY  [§  2. 

We  may  also  use  instead  of  (4)  or  (5), 

s" 


Equating  the  square  of  this  equation  to  the  preceding  one  gives 


which  gives  for  s  a  value  somewhat  exceeding  1000".  The 
general  rule  therefore  is  that,  in  using  any  angle  below  this 
limit,  the  sine,  tangent,  and  angle  may  be  used  indifferently. 

The  putting  of  1  for  the  cosine  of  a  small  angle  is  governed 
by  similar  considerations.  The  cosine  of  1000"  differs  from  1  by 
less  than  1  :  85  000.  Hence  if,  in  an  expression  of  the  form 

A  x  cos  s, 

an  error  0*000012^1  is  of  no  importance,  we  may  always  suppose 
coss  =  l    if   s"<1000".* 

3.   Unavoidable  errors  in  computation. 

We  cannot  determine  a  physical  quantity  with  mathematical 
exactness.  Measures  of  length,  weight,  volume,  and  every  other 
magnitude  are  liable  to  errors,  which  we  may  reduce  more  and 
more  by  laborious  attention  to  details,  but  can  never  absolutely 
eliminate.  Many  measures  may  be  in  error  by  their  thousandth 
part,  and  it  is  only  a  few  fundamental  quantities  which  we  can 
consider  as  known  within  their  millionth  part.  Even  were  a 
rigorous  determination  possible,  its  rigorous  expression  by  any 
system  of  numbers  would  not  be.  Our  systems  of  expressing 
quantities  numerically  by  an  infinite  series,  proceeding  according 
to  the  diminishing  powers  of  a  base,  is  the  best  that  can  be 
applied  in  practice.  In  our  traditional  system  of  numeration  the 
base  is  the  number  10.  Could  we  begin  anew,  the  number  12 
might  be  better;  but  this  is  impracticable.  In  a  decimal  ex- 
pression we  reduce  the  maximum  error  by  one-tenth  by  every 
figure  we  add,  but  can  only  approach  to  the  rigorous  value  of 
a  concrete  physical  quantity,  or  the  logarithm  of  a  number 
which  is  not  an  integral  power  or  root  of  10.  As  a  general 

*  Mention  may  here  be  made  of  the  almost  universal  practice  of  using  the  word 
*  '  arc  "  to  indicate  an  angle  expressed  in  degrees,  minutes,  or  seconds. 


§3.]  UNAVOIDABLE   ERRORS   IN  COMPUTATION  7 

rule,  there  is  no  use  in  adding  decimals  beyond  the  practically 
attainable  limit  of  accuracy. 

In  a  numerical  computation,  especially  with  logarithms,  one 
should  always  have  some  idea  of  the  degree  of  accuracy  attain- 
able or  desirable  ;  or,  to  speak  with  more  precision,  of  the 
magnitude  of  the  errors  to  which  the  data  and  results  are  liable. 
The  accuracy  of  a  result  is  limited  by  that  of  the  data  on  which 
it  depends,  so  that,  in  all  computations,  the  result  must  be 
affected  by  errors  arising  from  those  of  the  data,  no  matter  with 
what  precision  the  computation  is  made.  As  every  additional 
figure  used  in  computation  adds  to  the  labour,  the  first  question 
to  be  considered  by  the  computer  in  entering  upon  a  work  is  : 
How  many  figures  are  necessary  in  the  logarithms  in  order  that 
the  unavoidable  errors  of  the  result  may  not  be  increased  by  the 
errors  of  the  logarithms  ?  The  logarithmic  tables  in  ordinary  use 
range  from  three  to  seven  decimals,  and  the  question  of  the 
error  arising  from  the  decimals  following  the  last  being  omitted 
is  the  first  to  be  considered. 

Let  q  be  the  true  value  of  a  quantity  and  +  8  the  error  of  the 
value  we  derive,  so  that  q  +  S  is  the  value  we  reach  by  computa- 
tion. We  have  to  find  what  value  of  S  will  arise  from  using 
logarithms  from  which  the  decimals  after  a  certain  order  are 
dropped.  Treating  S  as  an  infinitesimal,  we  have,  for  the  error 
of  the  logarithm,  corresponding  to  the  error  8  of  q, 


log  (</  +  <?)-  log  <?  =  log   l+      =  M   ................  (6) 

M  being  the  modulus,  O434  29  — 

If  we  use  7i-place  logarithms,  the  value  of  the  unit  in  the  last 
figure  will  be  10  ~w.  In  taking  out  a  logarithm,  the  error  need 
be  only  a  fraction  of  this  unit  ;  but  in  adding  up  several,  it  may 
reach  or  exceed  the  unit.  Assuming  a  unit  error  in  the  last 
figure  of  log  q  we  shall  have 


io-n 

and  S  = 


8  INTRODUCTORY  [§  3. 

Assigning  to  n  the   successive  values  3  to  7  we  have  the 
corresponding  errors  of  q  as  follows  : 

3-place  logarithms,  8=  ±'0023g 
4-     „  „  ,,=  ±'00023g 


5-     „  „  ,,=  ±'000023g 

6-'    „  „  ,,  =  ±-0000023gf 

7-  ,  = 


(7) 


Using  round  numbers,  we  may  say  that  the  use  of  3-place 
logarithms  will  give  a  result  correct  to  the  400th  part  of  its 
amount,  4-place  logarithms  to  the  4000th  part,  and  so  on.  Con- 
versely, if  we  wish  a  result  correct  only  to  the  100th  part,  3-place 
logarithms  will  do  ;  to  the  1000th  part,  4-place  ;  to  the  1  :  10n 
part  we  should  use  n  +  1  decimals  in  the  logarithms. 

4.  The  preceding  rules  give  only  a  relation  between  the  errors 
of  a  logarithm  and  of  the  corresponding  number.  The  relation 
between  the  error  of  the  data  and  of  the  result  can  be  expressed 
thus  :  Let  the  given  data  be  u,  v,  w,  etc.  ;  the  quantity  to  be 
computed  p.  We  may  then  regard  p  as  a  function  of  u,  v,  w. 
If  we  put  Su,  Sv,  Sw,  for  the  errors  of  these  quantities,  the  error 
in  p  will  be 


If  the  values  of  the  derivatives  which  enter  into  this  expression 
are  large  numbers,  the  error  of  the  result  will  be  greater  than 
those  of  the  data  in  like  proportion. 

A  case  of  this  kind  occurs  in  determining  a  small  angle  by  its 
cosine,  or  one  near  90°  by  its  sine.  The  error  of  the  angle  may 
then  be  many  times  greater  than  that  of  the  function  by  which 
it  is  determined.  It  is,  therefore,  preferable  to  determine  an 
angle  by  its  tangent  when  this  can  be  done. 

In  ordinary  computation  a  common  case  of  this  kind  is  that  in 
which  the  result  comes  out  -as  a  difference  of  two  large  and  nearly 
equal  quantities,  or  as  the  quotient  of  two  such  differences,  or  of 
two  small  quantities.  In  such  a  case  more  logarithms  must  be 
used  in  computing  the  large  numbers,  or  the  small  numbers 
must  be  carried  to  a  higher  degree  of  precision,  than  would  be 
prescribed  by  the  rule. 


§5.]  DEEIYATIYES,   SPEEDS,  AND   UNITS  9 

5.   Derivatives,  speeds,  and  units. 

As  the  theoretical  study  of  the  differential  calculus  does  not 
suffice  for  its  practical  applications,  we  begin  with  some  remarks 
on  derivatives  and  the  units  in  terms  of  which  they  are  expressed. 

The  derivative  of  a  quantity  with  respect  to  the  time,  at  any 
moment,  represents  the  velocity  or  speed  of  increase  of  the 
quantity  at  that  moment.  If  the  increase  is  constant  the  speed 
is  found  by  dividing  the  increment,  whatever  it  may  be,  by  the 
time  necessary  for  that  increment  to  take  place.  If,  however, 
the  speed  of  increase  is  continually  varying,  it  is  at  any  moment 
the  quotient  of  the  infinitesimal  increase  of  the  quantity  by  the 
infinitesimal  time  required  for  that  increase.  Using  the  notation 
of  the  differential  calculus,  if  we  put  Q  for  the  quantity  and  8 
for  the  speed  we  have  7n 


In  the  present  work  we  use  the  more  compact  symbol  Dt  to 
express  the  derivative,  or  the  speed  of  increase  of  the  quantity 
following  it.  When  this  symbol  is  written  before  any  quantity 
Q  the  combination  DtQ  therefore  signifies  the  rate  of  increase  of 
Q  at  any  moment. 

The  next  question  concerns  the  units,  especially  of  time,  in 
which  the  speed  is  to  be  expressed.  The  fact  that  the  latter  is 
determined  by  an  increase  during  an  infinitesimal  time  sets  no 
limit  upon  the  length  of  the  unit  of  time  that  may  be  used. 
The  units  employed  in  astronomy  range  all  the  way  from  one 
second  to  100  years.  The  unit  of  speed  is  defined  as  that  speed 
which,  if  it  remained  constant  during  the  unit  of  time,  would 
produce  unit  increase  in  the  quantity  whose  speed  is  designated. 
In  ordinary  language  we  express  these  units  whenever  necessary, 
speaking,  for  example,  of  5  feet  per  second  or  15  degrees  per  hour 
or  20  seconds  per  century. 

The  relation  of  these  units  to  the  infinitesimals  by  which  a 
derivative  is  defined  needs  a  moment's  consideration.  If  we  say 
that  the  speed  of  increase  of  the  R.A.  of  a  star  is  300  s.  per 
century,  this  means  that  the  R.A.  would  increase  by  that  amount 
in  a  century  if  the  speed  remained  constant.  To  determine  or 


10  INTRODUCTORY  [§  5. 

express  the  derivative,  that  is  the  speed,  we  may  take,  instead  of 
an  infinitesimal  time  dt,  any  interval  during  which  the  speed 
may  practically  be  treated  as  constant.  In  the  motions  which 
affect  the  stars"  the  general  rule  is  that  the  speed  during  a  year 
varies  so  little  that  the  interval  adopted  may  be  a  year.  This 
has  led  to  the  use  of  symbols  in  a  double  meaning,  which  may 
sometimes  lead  to  confusion  if  the  difference  in  the  two  meanings 
is  not  understood.  For  example  the  symbol  m  is  used  to  express 
the  constant  part  of  the  precession  in  RA.  of  a  star  during  the 
year.  As  its  variation  during  any  one  year  is  too  minute  to  be 
considered,  the  annual  speed  of  the  precession  in  R.A  is  also 
indicated  by  the  symbol  m.  But  it  should  be  understood  that 
these  two  interpretations  are  different,  though  the  symbol,  and 
the  number  it  represents,  may  be  the  same. 

6.  Differential  relations  between  the  parts  of  a  spherical  triangle. 

In  a  large  class  of  astronomical  problems  the  given  quantities 
are  three  of  the  parts  of  a  spherical  triangle,  and  the  problem  is 
to  find  one  or  more  of  the  three  remaining  parts.  As  auxiliary 
to  this  problem  it  may  be  asked  what  changes  or  errors  in  the 
required  part  will  be  produced  by  given  small  changes,  or  errors 
treated  as  infinitesimal,  in  the  three  given  parts.  This  requires 
that  we  find  the  derivatives  of  the  required  parts  as  to  the  given 
parts,  the  latter  being  treated  as  independent  variables. 

There  are  three  independent  relations,  and  no  more,  between 
the  six  parts.  From  these  relations,  expressed  as  equations,  we 
may  eliminate  any  two  of  the  parts,  leaving  one  equation  between 
four  parts,  from  which  any  one  part  may  be  determined  in  terms 
of  the  other  three.  Let  such  an  equation  be  expressed  in  the 
form 

<j>(x,y,z,u)  =  0',  ...........................  (9) 

we  shall  then  have  by  differentiation 


From  this  equation  the  value  of  any  one  differential  in  terms 
of  the  three  others  may  be  found.     For  example 


§7.]  PARTS  OF  A  SPHERICAL  TRIANGLE  11 

d<j> 

dx  _  cly 
dy  ~  d(f> 

dx 

Since  there  are  15  combinations  of  4  parts  out  of  6,  we  may 
write  as  many  equations  of  this  form.  But,  only  three  of  these 
will  be  independent  of  each  other,  and  these  three  may  be  formed 
from  one  by  permutation  of  parts.  Let  us  take  the  fundamental 
equation  <£  =  cos  a  cos  b  +  sin  a  sin  b  cos  G  —  cos  c  =  0  ; 

we  shall  have    -r-  =  —  sin  a  cos  b  4-  cos  a  sin  b  cos  C 
da 

=  —  sin  c  cos  B,  ....................................  (11) 

-$.—  —  Cos  a  sin  b  +  sin  a  cos  b  cos  C 
do 

=  —  sinccos  A,  ...................................  (12) 


.............................................  (13) 

-^  =  —  sin  a  sin  b  sin  C.    .............................  (14) 

The  equation  between  the  four  differentials  may,  therefore,  be 
written  : 
sin  c  cos  Bda  +  sin  c  cos  Adb  —  sin  cdc  +  sin  <x.sin  b  sin  CdC  =  0.  (15) 

From  this  two  others  of  like  form  may  be  written  by  changing 
each  letter  into  the  one  next  following  in  alphabetical  order, 
A  and  a  following  C  and  c. 

The  forms  derived  from  these  for  practical  application  will  be 
found  in  Appendix  I. 

7.   Differential  spherical  trigonometry. 

In  using  the  differential  increments  of  the  parts  of  a  spherical 
triangle,  and  of  angles  and  arcs  generally  on  the  sphere,  a  great 
advantage  is  often  gained  by  treating  the  subject  geometrically. 
The  following  are  fundamental  theorems  at  the  base  of  the 
method  : 

(i)  An  infinitesimal  spherical  triangle  may  be  treated  as  a 
plane  triangle  when  infinitesimals  of  an  order  higher  than  the 
second  are  neglected. 


12  INTEODUCTOEY  [§  7. 

This  follows  from  the  fact  that  the  excess  of  the  sum  of  the 
angles  over  180°,  being  proportional  to  the  area  of  the  triangle, 
is  of  the  second  order ;  and  that  the  deviations  of  the  parts  from 
those  of  a  corresponding  plane  triangle  formed  by  the  projection 
of  the  spherical  triangle  on  a  tangent  plane  to  the  sphere  at  the 
place  of  the  triangle  are  of  the  second  order. 

It  must  be  noted,  however,  that  this  theorem  presupposes 
the  three  angles  to  be  finite  quantities,  and  is,  therefore,  not 
applicable  when  an  infinitesimal  angle  is  under  consideration. 
The  following  theorems  apply  to  cases  of  the  latter  kind  : 

(ii)  If  two  great  circles  intersect  at  C,  forming  the  infinitesi- 
mal angle  a,  their  distance  apart  at  a  distance  a  from  G  is 


FIG.  1. 

For,  supposing  AD=p  to  be  a  perpendicular  to  the  arc  CA, 
we  have 

sin  p  =  sin  a  sin  a, 

from  which  the  equation  follows  when  a  and  p  become  infini- 
tesimal. 

(iii)  An  arc  CA  cuts  the  transversal  arc  MN  at  the  point  A. 
If  this  arc  turns  on  G  through  the  infinitesimal  angle  a.  into  the 
position  CA',  the  corresponding  increment  of  the  angle  at  A 
will  be 


To  apply  the  usual  formulae  of  spherical  trigonometry  to  this 
case  we  put  B  for  the  interior  angle  adjacent  to  A'y  and  letter 
the  remaining  parts  of  the  triangle  ABC  accordingly.  Then 

A'  -A  =  180°  -(A  +  B), 


§7.]  DIFFERENTIAL   SPHERICAL  TRIGONOMETRY  13 

From  the  fundamental  equation 

sin  A  cos  B  4-  cos  A  sin  B  cos  c  =  cos  b  sin  C, 
we  find,  when  c  and  C  are  infinitesimals, 

sin  ( A  +  B)  =  cos  b  sin  (7  =  sin  a  cos  6. 

In  this  case  the  part  C  reduces  to  a,  and  the  part  b  to  the  arc 
CA.  Hence,  when  G  becomes  infinitesimal  we  have  the  equation 
enunciated. 


NOTES  AND   REFERENCES. 

A  prime  requisite  to  the  astronomical  computer  is  a  set  of  the  most 
convenient  logarithmic  and  other  tables.  The  following  are  of  this  class  : 

Seven-place  logarithms. 

BRUHNS,  New  Manual  of  Logarithms,  Tauchnitz,  Leipzig. 

ZECH,  Additions-  und  Subtractions- Log  arithmen,  Hirzel,  Leipzig. 

The  purpose  of  these  last  named  tables  is  to  find  the  logarithm  of  the  sum 
or  difference  of  two  numbers  given  by  their  logarithms,  without  the  labour 
of  taking  out  the  natural  numbers  and  adding  them. 

Six-place  logarithms. 

BREMIKER,  Logarithmisch-trigonometrische  Tafeln,  Berlin,  Nicolaische  Ver- 
lags-Buchhandlung. 

This  edition  contains  also  addition  and  subtraction  logarithms. 

Five-place  logarithms. 

GAUSS,  Fiinfstellige  vollstdndige  Logarithmische  und  Trigonometrische  Tafeln, 
Halle,  Yerlag  von  Eugen  Strien. 

HUSSEY,  Logarithmic  and  Other  Mathematical  Tables,  Ann  Arbor,  Mich., 
the  Register  Publishing  Co. 

NEWCOMB,  Five-place  Logarithmic  Tables,  New  York,  Henry  Holt  &  Co. 

BECKER,  Logarithmisch-trigonometrisches  Handbuch  auf  5  Decinialen, 
Tauchnitz,  Leipzig. 

The  tables  of  Becker,  of  Gauss,  and  of  Newcomb  contain  addition  and 
subtraction  logarithms,  and  other  useful  tables.  The  introduction  to 
Newcomb's  tables  contains  hints  on  the  art  of  astronomical  computation  to 
facilitate  the  training  of  computers. 

Four-place  tables. 

GAUSS,   Vierstellige  Logarithmisch-trigonometrische  Handtafeln. 


14  INTRODUCTORY 

Three-place  tables. 

NEWCOMB,  3-  and  4:-place  Logarithmic  and  Trigonometric  Tables,  Appendix 
to  Elements  of  Trigonometry,  New  York,  Henry  Holt  &  Co. ;  also  separately, 
Washington,  Lowdermilk  &  Co. 

Tables  of  special  kinds. 

Besides  the  above  tables  which  may  be  described  as  in  regular  form,  the 
following  are  useful  for  special  purposes : 

GRAVELIUS,  Funfstellige  Logarithmisch-trigonometrische  Tafeln,  Berlin, 
Reimer. 

In  these  tables  the  degree  is  not  divided  into  minutes  and  seconds  but 
into  hundredths,  which  is  more  convenient  when  a  translation  into  minutes 
and  seconds  is  not  necessary. 

For  numbers  with  not  more  than  three  digits,  when  the  logarithm  of  the 
product  is  not  required,  and  when  only  two  factors  enter,  a  multiplication 
table  may  be  used  with  more  convenience  than  logarithms.  The  most 
extended  set  of  tables  of  this  sort  is  : 

CRELLE-BREMIKER,  Rechentafeln,  Berlin,  Reimer. 

TAMBORREL,  Tobias  de  Multiplicar,  Mexico,  Mariano*  Nava,  is  a  much  more 
compact  book  than  Crelle's,  and  may  serve  the  same  purpose. 

LECOY  ET  CLAUDEL,  Comptes  faits,  Paris,  gives  all  products  of  three  figures 
by  two. 

Among  miscellaneous  tables  for  astronomical  uses  in  general : 

BAUSCHINGER,  Tafeln  zur  Theoretiscken  Astronomic,  is  a  well  prepared  and 
most  useful  work. 

Astronomical  ephemerides. 

In  the  same  class  with  the  preceding  may  be  placed  the  National  Astro- 
nomical Ephemerides,  published  by  the  governments  of  the  United  States, 
Germany,  England,  France,  and  Spain  under  the  respective  titles  : 

The  American  Ephemeris  and  Nautical  Almanac,  Bureau  of  Equipment, 
Navy  Department,  Washington. 

The  Nautical  Almanac  and  Astronomical  Ephemeris,  published  by  the  order 
of  the  Lords  Commissioners  of  the  Admiralty,  London. 

Astronomisches  Jahrbuch,  Berlin. 

Connaissance  des  Temps  ou  des  Mouvements  celestes,  Paris,  Bureau  des 
Longitudes. 

Almanaque  Nautico,  San  Fernando,  Spain. 

These  publications  contain,  for  each  year,  tables  of  the  varying  quantities 
relating  to  the  celestial  motions.  They  are  referred  to  collectively  in  the 
present  work  under  the  title  of  the  Astronomical  Ephemeris. 


CHAPTEE   II. 

OF  DIFFERENCES,   INTERPOLATION,   AND 
DEVELOPMENT. 

8.  In  astronomical  tables  and  ephemerides  the  values  of  certain 
quantities  are  given  for  special  equidistant  epochs  or  values  of 
an  argument:  say  noon  of  every  day,  the  beginning  of  every 
year,  or  every  minute  of  the  quadrant.  When  the  values  are 
required  for  an  intermediate  epoch,  or  value  of  the  argument, 
the  process  of  interpolation  is  necessary.  There  are  various 
methods  of  applying  this  process,  according  to  the  greater  or  less 
complexity  of  the  law  of  change  of  the  tabular  quantities  to  be 
found. 

We  first  call  to  mind  that,  by  the  rate  of  variation  of  a 
quantity  at  a  given  moment,  we  mean  the  derivative  of  the 
quantity  as  to  the  time  at  that  moment;  that  is  to  say,  the 
change  which  the  quantity  would  undergo  in  a  unit  of  time  if 
its  rate  of  change  remained  constant.  The  unit  may  be  a  second, 
minute,  hour,  day  or  year,  or  even  a  century.  The  rate  may  be 
called  the  variation  simply.  The  simplest  cases  of  interpolation 
are  the  two  following : 

CASE  I.  The  variation  constant.  This  is  so  simple  a  case  as 
to  need  no  explanation.  To  effect  an  interpolation  we  have  only 
to  multiply  the  variation  by  the  elapsed  time  and  add  the  pro- 
duct to  the  value  given  in  the  table.  We  must  of  course  take 
care  that  the  unit  of  time  which  we  use  corresponds  to  that  for 
which  the  variation  is  given. 


16  DIFFERENCES,   INTERPOLATION,   DEVELOPMENT      [§  8. 

CASE  II.  When  the  variation  itself  changes  uniformly  with 
the  time.  We  may  treat  this  case  by  the  infinitesimal  calculus 
as  follows  : 

Call  u  the  given  quantity,  and  let  a  and  b  be  constants  ;  the 
variation  of  u  is,  by  hypothesis,  of  the  form 

du 

di 

/ 

From  this  we  derive  by  integration 


^  .............  ........... 

or  u  =  uQ  +  (a+^bt)t  * 

u0  being  the  value  at  the  time  from  which  t  is  reckoned. 

The  second  form  is  commonly  the  most  convenient  one  to  use, 
and  may  be  correctly  arrived  at  in  this  way:  By  hypothesis  we 
have 

At  time  t  =  0  ;  variation  =  a. 


It  will  be  seen  that  the  variation  which  we  found  in  formula 
(1)  is  the  half  sum  of  these  variations,  that  is,  the  value 
corresponding  to  the  middle  of  the  interval  over  which  we 
interpolated. 

As  an  example,  let  us  take  from  the  Ephemeris  the 
Right  Ascension  of  the  Moon  at  two  consecutive  hours  of 
Greenwich  mean  time,  on  1908,  June  13,  which  we  find  to 
be  as  follows: 

TT  -r>   .  Variation  for 

Hour-  RA"  1  minute. 

h.    m.         8.  s. 

1  16  27  35-43      2'4280 

2  16  30  .  1-31      2-4347 

Let  it  be  required  to  interpolate  to  the  time  1  h.  36  m.  Since 
the  variation  is  constantly  increasing,  it  is  clear  that  if  we  used 
the  variation  at  1  h.  the  resulting  difference  would  be  too  small  ; 
and  if  we  used  the  variation  at  the  time  to  which  we  interpolate, 


§  9.]  DIFFERENCES  OF  VARIOUS   ORDERS  17 

namely,  1  h.  36  m.,  it  would  be  too  great.  What  we  must  do  is 
to  take  the  variation  at  the  moment  which  is  half-way  between 
the  epochs,  namely,  1  h.  18  m.  or  1*3  h.  This  we  find  by  simple 
interpolation  to  be  2'4280s.  -f  0*0067  s.  X  O3  =  2-4300  s.  Multiply- 
ing this  rate  of  change  per  minute  by  the  36  minutes  which  have 
elapsed,  we  find  the  interpolated  value  to  be 


16     27     35-43  +  1     27*48  =  16     29     2*91 

In  many  tables  and  ephemerides,  what  is  given  is  not  the 
derivative,  or  variation  per  unit  of  time,  but  the  difference 
between  two  consecutive  values  of  the  quantity,  which  is  found 
by  subtracting  each  value  from  the  one  which  follows  it.  If 
we  do  this  with  the  R.A.'s  in  the  lunar  ephemeris,  we  shall  find 
that  the  differences  vary  nearly  uniformly  from  hour  to  hour, 
and  a  little  consideration  will  show  that  in  this  case  they  express 
the  hourly  variation  at  each  half  hour.  The  variation  corre- 
sponding to  the  middle  of  the  interval  over  which  we  interpolate 
may  then  be  found  by  interpolating  to  this  moment  from  the 
variations  at  the  half  hour  preceding  and  following. 

9.    Differences  of  various  orders. 

As  a  general  rule  the  quantities  found  in  the  ephemeris  are 
given  for  intervals  so  short  that  the  preceding  methpds  of  inter- 
polation will  suffice  to  give  an  accurate  value  of  the  quantity 
at  any  moment.  But  cases  continually  arise  in  astronomical 
practice  in  which  the  variation  itself  changes  widely  between 
two  epochs.  This  more  general  case  requires  us  to  begin  by 
pointing  out  the  method  of  testing  the  accuracy  of  numbers 
by  successive  differences,  defined  as  follows  : 

When  we  have  several  successive  values  of  a  varying  quantity, 
the  excess  of  each  value  over  that  preceding  is  called  a  first 
difference,  or  difference  of  the  first  order. 

The  excess  of  each  first  difference  over  the  preceding  one  is 
called  a  second  difference,  or  difference  of  the  second  order. 

Continuing  this  process  of  subtraction  we  have  third  differ- 
ences, fourth  differences,  etc. 

N.S.A.  B 


18  DIFFERENCES,   INTERPOLATION,   DEVELOPMENT      [§9. 

The  successive  differences   are   generally  designated   by  the 
A',  A",  A",   A* 


symbols 


The  best  form  for  writing  differences  is  that  shown  in  the 
scheme  (B)  following.  The  first  column  contains  only  a  series  of 
indices  which  serve  for  the  numbering  of  the  individual  differ- 
ences. The  next  column  gives  the  successive  values  of  the 
function,  which  we  call  u.  We  then  have 


^A;_A;,    et, 


_A 


etc. 


We  arrange  these  quantities  as  follows  : 


..(A) 


0     u0 

i  u,  -    A;- 

-  A?    -   A'if 

2     wj    —       A;'    —       AJ7 

V                                                                       "!?•                                                                         0 

3     ^3               Ag               A3V     — 

A;          A^         A^ 

4     ?£4     —       A!'    —       A7    — 

AO             Ag"           A! 

5     -?/                  A"                Alv 
o     -zt5                n5                A5 

e  «,6  -5   A;-  -?   A? 

7  u7         A;' 

o 

(B) 


It  will  be  seen  that  each  difference  is  written  on  the  line 
between  the  two  numbers  to  whose  difference  it  is  equal,  and  is 


§io.] 


DIFFERENCES  OF  VAEIOUS  ORDERS 


19 


distinguished  by  a  suffix  equal  to  half  the  sum  of  their  suffixes. 
Thus,  like  suffixes  are  on  a  horizontal  line;  differences  of  even 
order  all  have  integral  suffixes ;  those  of  odd  order  fractional 
ones. 

As  an  example  we  take  the  moon's  longitude  for  Greenwich 
noon  and  midnight  of  the  first  few  days  of  1895  and  differ- 
ence it. 


1895. 

Jan.         Longitude.  A' 

1-0    33<)°36'53"'6 

5°  55'  50"7 
1-5    345   32  44  '3  1'55"'9 

5  57  46  -6  38"-4 

2-0    351   30  30  -9  2  34  '3  +  2"-l 

6  0  20  -9  40  -5  -2"-0 
2-5    357  30  51  -8                         3  14  '8               +0  -1 

6     3  35  7  40  -6  +0  -5 

3-0        3  34  27  '5  3  55  -4  +0  '6 

6     7  31  -1 
3-5        9  41  58  -6 

6  12    77  39  '9 

4-0      15   54    6  -3  5  16  '5 

6  17  24  -2 
4-5      22   11  30  -5 


A"          A" 


41  '2  -1  -9 

4  36  -6  -1  '3 


(c) 


10.   Detecting  errors  by  differencing. 

One  of  the  most  valuable  applications  of  differencing  is  to  the 
detection  of  isolated  errors  in  the  values  of  the  functions  whose 
differences  are  taken.  Suppose  that  one  of  the  values  of  u  is 
affected  by  the  error  e,  so  that  the  table,  instead  of  giving  the 
value  u,  gives 


while  all  the  other  numbers  are  correct.  We  then  readily 
conclude  that  the  first  differences  before  and  after  this  quantity 
will  be  affected  by  the  respective  errors  +e  and  —e.  The  second 
differences  will  be  affected  by  the  errors  +e,  —2e,  and  +e. 
Continuing  the  process  we  find  the  resulting  errors  of  the 
successive  differences  to  be  as  follows  : 


20 


DIFFERENCES,   INTERPOLATION,   DEVELOPMENT    [§  10. 

ofAv 


of  A'    of  A"    of  A'"    ofAiv 
—          0          —          0 


—  0 


-2e        —        +6e         — 


106 
-10* 


0 

0 
0 

0 
0  — 


—  0  — 


H-e 


—  0  — 


0  — 


0 


•(2) 


We  see  that  an  error  in  any  one  original  quantity  will  be 
increased  tenfold  when  carried  out  to  the  fifth  difference,  and 
can  in  all  ordinary  cases  be  detected,  provided  the  adjacent 
quantities  are  correct. 

The  general  expression  for  the  coefficients  of  e  in  the  errors  of 
the  nth  differences  is  the  same  as  that  of  the  coefficients  which 
enter  into  the  binomial  theorem,  namely 


1-2-3-...S 

where  s  takes  the  successive  values  1,  2,  3,  ...  n.  In  applying 
this  test  it  must  be  remembered  that  all  the  quantities  we 
ordinarily  obtain  in  astronomical  computations  are  necessarily 
affected  by  the  errors  of  the  omitted  decimals,  which  errors  will 
shew  themselves  by  the  process  of  differencing. 

How  far  it  is  necessary  to  carry  the  differences  will  depend 
upon  the  rapidity  with  which  they  converge.  If  the  given 
numbers  are  mathematically  exact,  the  differences  will,  if  the 
quantities  are  given  for  values  of  the  argument  small  enough  to 
be  used  in  interpolating,  continually  and  rapidly  diminish,  so 
that,  after  a  certain  order,  commonly  not  greater  than  the  fifth 
or  sixth,  they  become  insensible.  But  the  differences  of  the 
errors  arising  from  omitted  decimals  will,  as  just  shown,  go  on 


§  11.]  DETECTING  ERRORS   BY   DIFFERENCING  21 

increasing  with  every  order,  and  so  will  ultimately  form  the 
largest  part  of  the  column  of  differences.  When  this  is  the  case 
the  columns  of  differences  will  become  irregular,  the  +  and  — 
signs  generally  alternating. 

Other  points  to  be  borne  in  mind  are  these : 

a.  If  errors  are  numerous  but  accidental,  the  fact  that  they 
exist  will  be  shown  by  the  differences,  but  it  may  be  impossible 
to  determine  what  numbers  are  wrong;  whereas  this  is  easy  in 
the  case  of  a  single  isolated  error. 

/3.  A  systematic  error,  i.e.,  one  which  runs  through  all  the 
numbers  and  follows  any  law  whatever,  cannot  be  detected  by 
differencing. 

11.  Use  of  higher  orders  of  differences  in  interpolation. 

There  are  two  applications  of  the  method  of  interpolation  by 
differences. 

(i)  When,  from  several  values  of  a  variable  quantity,  given  in 
tabular  form,  it  is  desired  to  find  some  intermediate  value  for 
one  or  more  special  values  of  the  argument. 

(ii)  When  it  is  desired  to  make  a  table  in  which  the  intervals 
shall  be  smaller  than  those  of  the  quantities  originally  computed. 
For  example,  the  position  of  a  planet  may  be  computed,  in  the 
first  place,  for  every  fourth,  fifth,  or  tenth  day ;  and  it  may  then 
be  required  to  form  an  ephemeris  for  all  the  omitted  days  by 
interpolation.  In  the  second  application  the  same  general 
formulae  are  used  as  in  the  first ;  we  shall  therefore  give  a  brief 
summary  of  them. 

First  application.  As  before,  we  put  u  for  the  variable 
quantity  for  which  we  have  computed  the  values  for  a  number 
of  equidistant  epochs ;  we  suppose  the  successive  differences  of  u 
formed  so  far  as  necessary,  and  we  call 

UQ  and  Uj_ 

the  two  consecutive  values  of  u  between  which  we  wish  to 
interpolate  a  new  value. 

It  is  a  known  theorem  of  algebra  that  the  nth  value  of  u 
following  u0,  which  is  in  fact  un,  is  given  in  terms  of  UQ,  and  of 


22  DIFFERENCES,  INTERPOLATION,  DEVELOPMENT      [§  11. 

the  successive  differences  in  a  diagonal  line  descending  from  u0 
by  the  equation 

A,  ,  n(n—  !).„     ™(n  —  I)(TI  —  2)  .,„  ,  /ox 

i<n  =  u0+?iA;  +    V1>2   ^  +  --  j-^j  --  'AJ'+.  ......  .(3) 

the  coefficients  being  those  of  the  binomial  theorem. 

The  fundamental  hypothesis  of  interpolation  is  that  this 
equation,  which  is  rigorous  only  when  n  is  a  positive  integer, 
will  also  give  the  value  of  u  when  n  is  a  fraction.  This  hypo- 
thesis, though  not  necessarily  true,  and  failing  entirely  in  cases 
when  the  law  of  change  in  u  is  not  shown  by  the  differences,  is 
quite  safe  in  practice  if  we  compute  the  values  of  u  for  intervals 
so  small  that  the  orders  of  differences  are  convergent  both  for 
these  and  for  all  shorter  intervals.  Let  us  put 

TQ  ;  T:  ;  the  two  times  for  which  the  values  u0  and  uv  between 
which  we  interpolate  a  new  value,  are  computed. 

T,  the  time  for  which  we  wish  to  interpolate.     Then 


will  be  the  time  after  T0,  expressed  in  terms  of  the  interval  of 
computation  as  the  unit.  For  example,  if  this  interval  were  two 
days,  and  we  wished  to  find  the  value  of  u  for  a  moment  8  hours 
after  one  of  the  times  of  computation,  we  should  put 

£  =  8-^-48=0-1666.... 

Evidently  when  t  is  an  integer  it  will  correspond  to  n,  as 
already  defined,  so  that  the  equation  (3)  becomes 

(4) 


This  is  Newton's  formula  of  interpolation,  and  forms  the  basis 
of  all  the  other  formulae  in  common  use. 

12.  Transformations  of  the  formula  of  interpolation. 

In  Newton's  formula  each  successive  difference  which  enters 
is  taken  half  a  line  below  the  preceding  one.  The  series  is, 
however,  more  convergent  when  the  differences  of  alternate 
orders  are  taken  on  the  same  horizontal  line.  The  transforma- 
tions to  effect  this  will  now  be  shown. 


§12.]  FORMULA  OF  INTERPOLATION  23 

In  all  cases  in  which  it  is  worth  while  to  interpolate,  the 
differences  beyond  the  fifth  will  not  affect  the  result.  We  shall 
therefore  suppose  the  differences  of  the  sixth  and  all  following 
orders  to  vanish,  which  amounts  to  supposing  those  of  the  fifth 
order  constant. 

We  first  consider  the  form  in  which  the  original  differences  on 
alternate  lines  are  used.  Let  us  express  ut  in  terms  of  the 
quantities  shown  in  the  following  scheme  : 

A»       A»v   - 
-  A;   -  AJ   - 

We  express  the  differences  in  (4)  in  terms  of  these  as  follows  : 


+AT 


Making  these  substitutions  in  (4)  and  putting  Avi  =  Avii  =  0,  we 
find,  by  reducing  and  collecting  the  coefficients  of  the  differences 
in  (b), 


I    V"    I    J-/t/V/        -*-/\"        -^/  Aivi    V      '       /\      '    •VV''  /\  7AY 


or  u,  — un  = 


1.2.3.4 


.(5) 


1.2.3.4.5 

Let   us   next   take   the   differences   of   odd   orders    one    line 
higher,  thus:       _     ^     _     ^     _     ^ 

We  have  A ,       A , 


-i 


24  DIFFERENCES,  INTERPOLATION,  DEVELOPMENT     [§  12. 

Making  these  substitutions  in  (5),  we  find 


,     (6) 

T72.3.4.5  "-• 
Now    let    us    make    a    third   transformation   by   using   the 
differences  next  below  those  of  (c),  thus : 

) 
-  ^  -  Ai       AI <<>•> 

11  A"  Aiv 

^  l±i  Hj 

If,  in  (G),  we  substitute  these  differences  for  those  there  written, 
we  shall  have  the  value  of  ut+l  —  uv  that  is,  putting 


we  shall  have  the  value  of  u?  —  u^  =  u^  —  UQ  —  A  \  .  Substituting  in 
(6)  for  t  its  value  if  —  1  and  the  differences  (d),  and  then  dropping 
the  accent,  we  find 


(7) 


1.2.3.4.5 


The  formulae  (5),  (6),  and  (7)  have  the  advantage  that  the 
differences  of  each  alternate  order  are  taken  from  the  same 
horizontal  line.  But  a  yet  farther  transformation  is  necessary 
to  reduce  the  equations  to  the  best  form  for  practical  use. 

Referring  to  the  scheme  (B)  it  will  be  seen  that  there  are  no 
values  of  A',  A'",  etc.,  with  entire  suffixes,  and  no  values  of  A", 
Aiv,  etc.,  with  fractional  suffixes,  but  that  the  places  where  these 
values  might  go  are  left  blank.  Now,  we  may  imagine  that 
these  blanks  are  filled  by  quantities  given  by  the  general  equation 


that  is,  in  each  blank  space  we  may  imagine  written  the  half 
sum  of  the  A's  above  and  below  it,  and  we  designate  these 
half  sums  by  A  with  half  the  sum  of  the  suffixes  attached.  We 
use  this  notation  in  what  follows. 


§  14.]  STIRLING'S   FOEMULA  OF  INTERPOLATION  .      25 

13.  Stirling's  formula  of  interpolation. 

In  this  formula  the  following  scheme  is  used  : 

u0,  A;,  AO,  AO',  A0V,  AJ,  etc. 

AO,  AO",  AO,  etc.,  being  defined  as  just  described.     We  derive  the 
formula  by  taking  the  half  sum  of  (5)  and  (6),  which  gives 


/Q\ 

' 


3°        1.2.3.4.5 

which  may  be  used  equally  well  for  either  positive  or  negative 
values  of  t 

14.  Bessel's  formula  of  interpolation. 

The  scheme  of  A's  used  in  this  formula  is 


A;     AT    AJ    A-    AJ   ...................  (e) 

u,     ----      -} 

The  formula  is  found  by  taking  the  half  sum  of  (5)  and  (7) 
which  is  : 


t(V-\)(t-V)          t(t*-l)(t-2)(t-l)        '  ( 

1.2.3.4     ^         1.2.3.4.5 
which  is  Bessel's  formula  in  its  usual  shape. 

The  second  member  of  this  is  less  simple  than  that  of  Stirling's 
formula,  where  the  differences  of  odd  order  have  as  coefficients 
only  odd  powers  of  t,  and  those  of  even  order  only  even  powers. 
But  we  may  make  it  more  symmetrical  by  filling  the  blank 
between  UQ  and  u^  in  the  scheme  (e)  by  K'^o+'^i)  =  u'±>  an(^  count- 
ing t  (considered  as  the  time)  from  the  corresponding  moment, 
which  we  may  do  by  putting 


First  we  shall  have  for  substitution  in  (10) 

(11) 


26  DIFFERENCES,   INTERPOLATION,   DEVELOPMENT    [§  14. 

Then  by  substitution  in  (10) 


1.2.3.4  1.2.3.4.5 


,  (12) 
+ete. 


which  is  now  as  symmetrical  as  Stirling's  formula. 

When  one  or  more  isolated  values  are  to  be  interpolated,  either 
of  the  formulae  (8),  (9),  or  (10)  may  be  used  with  nearly  equal 
convenience.  In  practice  it  will  often  be  convenient  to  use  (7), 
factoring  it  thus  : 


The  most  common  application  of  interpolation  is  the  second 
•described  in  §  11.  Such  an  interpolation  is  said  to  be  to  halves, 
to  thirds,  to  fifths,  etc.,  according  as  the  new  intervals  are  one 
half,  one  third,  one  fifth,  etc.,  of  the  original  ones.  The  most 
expeditious  way  of  executing  such  an  interpolation  is  to  compute 
the  first  differences  of  the  interpolated  series,  and  then  find  the 
required  quantities  in  succession  by  adding  these  differences. 

The  following  examples  of  the  way  of  practically  executing 
the  work  are  mostly  from  the  Introduction  to  the  author's  tables 
of  five-place  logarithms. 

15.  Interpolation  to  halves. 

It  is  required,  from  the  logarithms  of  310,  320,  330  ...  360  to 
find  those  of  315,  325  ...  355. 

Here  the  required  quantities  depending  upon  arguments  half 
way  between  the  given  ones  ;  we  have  t  =  J,  and  the  values  of  the 
JBesselian  coefficients,  so  far  as  wanted,  are 

t(t-l)         1 
2  8' 


2)_  3 
24  ~128* 


§15.]  INTERPOLATION  TO  HALVES  27 

The  subsequent  terms  are  neglected,  being  insensible.  So,  if 
we  put  a0  and  ax  for  any  consecutive  two  of  the  numbers  300, 
310,  etc.,  we  have 


3    Af  +  AyV 

2"*     8        2       "128        2       A 


where  we  put  A  'i  for  the  difference  between  log  a0  and  log  ar 

These  two  formulae  are  two  expressions  for  the  same  quantity, 
because  a0  +  5  =  «x  —  5.  They  are  both  used  in  such  a  way  as  to 
provide  a  check  upon  the  accuracy  of  the  work.  For  this  purpose 
we  compute  the  two  quantities 


...(15) 


The  most  convenient  and  expeditious  way  of  doing  the  work 
is  shown  in  the  accompanying  table,  where  we  give  every  figure 
which  it  is  necessary  to  write.  The  letters  (a),  (6)  . . .  (i)  at  the 
tops  of  the  columns  show  the  order  in  which  the  columns  are 
written. 


(h)  (g)  (f)  (e)  (c)  (d) 


No.  Log.  Diff.  ±A'  go  2  A' 

310       2-491 36 

695 
315  -49831  689-5  -5'5  -44  1379 

684 
320  -50515  -43 

673 
325  -51188  668-0  -51  -41  1336 

663 
330  -51851  -39 

653 
335  -52504  648-5  -4-8  -38  1297 

644 


28  DIFFERENCES,   INTERPOLATION,   DEVELOPMENT     [§  15. 

(c)  (d) 

A'  A" 

-38 
1259 

-36 
616 

611-5       -4-3         -34     1223 
607 
360       255630 

We  compute  the  column  (e)  by  the  formula 


(a) 
No. 

340 

(6) 
(0 

Log. 
•53148 

(A) 
Diff. 

(a)             (/) 
IA,        !AO  +  A" 

(e) 

AO  +  AI 

24          82 

2 

634 

345 

•537  82 

629-5        -4-6 

-37 

625 

350 

•54407 

616 

355 

•55023 

611-5       -4-3 

-34 

607 

the  set  of  suffixes  0,  1  and  J  being  applied  in  succession  to  each 
set  of  differences  which  enter  into  the  computation. 

This  mode  of  computing  the  half  sum  of  two  numbers  which 
are  nearly  equal  is  easier  than  adding  and  dividing  by  2. 

In  the  next  two  columns  to  the  left,  the  sixth  place  of  decimals 
is  added  in  order  that  the  errors  may  not  accumulate  by  addition. 
This  precaution  should  always  be  taken  when  the  interpolated 
quantities  are  required  to  be  as  accurate  as  the  given  ones. 

The  fourth  column  from  the  right  is  formed  by  adding  and 
subtracting  the  numbers  of  the  second  and  third  columns 
according  to  the  formula  (15).  The  additional  figure  is  now 
dropped,  because  no  longer  necessary  for  accuracy.  The  numbers 
thus  formed  are  the  first  differences  of  the  series  of  logarithms 
between  the  given  ones,  as  will  be  seen  by  equation  (15). 

We  write  the  first  logarithm  of  the  series,  namely, 

log  310  =2-491  36, 

and  then  form  the  subsequent  ones  by  continual  addition  of  the 
differences,  thus : 

Iog315  =  log  310+695; 

log  320  =  log  315 +  684; 

log  325  =  log  320 +673; 
etc.,       etc.,       etc. 


§  16.]  INTERPOLATION  TO  THIRDS  29 

If  the  work  is  correct,  the  alternate  logarithms  will  agree 
with  the  given  ones  in  the  former  table. 

The  continuance  of  the  above  process  for  a  few  more  numbers, 
say  up  to  450,  is  recommended  to  the  student  as  an  exercise. 

16.  Interpolation  to  thirds. 

Let  the  value  of  a  quantity  be  given  for  every  third  day,  and 
the  value  for  every  day  be  required.  By  putting  t  =  J-  and  apply- 
ing Bessel's  formula  to  each  successive  given  quantity,  we  shall 
have  the  value  for  each  day  following  one  of  those  given,  and  by 
putting  t=^  we  shall  have  values  for  the  second  day  following, 
which  will  complete  the  series.  But  the  interpolation  can  be 
executed  by  a  much  more  expeditious  process,  which  consists  in 
computing  the  middle  difference  of  the  interpolated  quantities 
and  finding  the  intermediate  differences  by  a  secondary  inter- 
polation. 

Let  us  put 

/0,/3,/6,  etc.,  the  given  series  of  quantities; 

fo>  fi>  fz>  /3>  /4'  e^c-'  the  required  interpolated  series ; 

A',  A",  etc.,  the  first  differences,  second  differences,  etc.,  of 

the  given  series ; 
8',  3",  etc.,  the  first  differences  second  differences,  etc.,  of 

the  interpolated  series. 
We  may  then  put 

(in  the  given  series) ; 


(in  the  interpolated  series). 


We  shall  then  have      S\  +  &  -f  &  =  A  i  . 

>          >          T  T 

The    value   of  /i—  /0  =  <Si    is   given   by   putting   t  =  ^  in    the 
Besselian  formula  (10).     Thus  we  find 


i    ,„    5 

9       2~~ 


30          DIFFERENCES,   INTERPOLATION,   DEVELOPMENT     [§  16* 

Putting  £  =  -§-,  we  have  the  value  of  /2—  /0,  that  is,  of  < 
Thus  we  find 


Subtracting  these  expressions  from  each  other,  we  have 


which  is  easily  computed  in  the  form 


We  see  that  the  computation  of  $,,  the  middle  difference  of 
the  interpolated  quantities,  is  much  simpler  than  that  of  ft.  It 
will  therefore  facilitate  the  work  to  compute  only  these  middle 
differences,  and  to  find  the  others  by  interpolation. 

This  process  is  again  facilitated,  in  case  the  second  differences 
are  considerable,  by  first  computing  the  second  differences  of  the 
interpolated  series  on  the  same  plan.  The  formulae  for  this 
purpose  are  derived  as  follows  : 

Let  us  put  <%=/4-/8- 

The  second  difference  of  which  we  desire  the  value  is  then 


The  value  of  ft  is  given  by  the  equation 


and  the  value  of  ft.  is  found  from  that  of  S\  by  simply  increasing 
the  indices  of  the  differences  by  unity,  because  it  belongs  to  the 
next  lower  line. 
We  thus  find 


9       2 

IA;+A; 


243        2  1458 


§  16.]  INTERPOLATION  TO  THIRDS  31 

Then  by  subtraction, 


^ 


243  2  1458 

Reducing  the  first  of  these  terms,  we  have 

A;-A^=A;'. 

For  the  second  term,       AQ  =  A"  —  A  7  ; 


whence  AJ  +  AJ  =  2  A"  +  AT  -  AT  =  2  AI  +  A!V, 

and  .         ^1±AL2A»  +  1A-,       "'" 

For  the  third  term,        AT  -  AT  =  A{7. 

For  the  fourth  term,  dropping  the  terms  in  Avi  as  too  small 
in  practice,  we  may  put 


1 A 


The  difference  of  the  fifth  terms  may  also  be  dropped,  because 
they  contain  only  sixth  differences. 

Making  these  substitutions  in  the  value  of  $J,  we  find 


> (17) 


By  this  formula  we  may  compute  every  third  value  of  S",  and 
then  interpolate  the  intermediate  values.  By  means  of  these 
values  we  find  by  addition  the  intermediate  values  of  S',  of 
which  every  third  value  has  been  computed  by  formula  (16). 
Then,  by  continually  adding  the  values  of  S',  we  find  those  of 
the  function  /. 


32  DIFFERENCES,   INTERPOLATION,   DEVELOPMENT     [§  16. 

As  an  example  of  the  work,  we  give  the  following  values  of 
the  sun's  decimation  for  every  third  day  of  part  of  July  1886, 
for  Greenwich  mean  noon  : 

Date.  0's  Dec.  A'  A"  A'" 

1886.  o       /  ,  „  ,    • 

July    3         22  57  37-5 

-16  28-3 
6         22  41     9-2  -212-4 

-20     0-7  +4-5 

9         22  21     8-5  -207-9 

-23  28-6  +4-5 

12         21  57  39-9  -203-4 

-26  52-0  +57 

15         21  30  47-9  -197-7 

-30     9-7 
18         21     0  38-2 

The  values  of  Aiv  are  too  small  to  have  any  influence. 

The  whole  work  of  interpolation  is  shown  in  the  following 
table,  where,  as  before,  the  right-hand  column  is  that  first 
computed,  and  gives  the  value  of  A'— -^A'"  according  to 
formula  (16): 

Date.  0's  Dec.  5'  5"  A'-^'" 

1886.  o       /          //  ,n  n 

July    6        22  41    92  -23'60 

-6  16-86 

7  22  34  52-4  -23'43 

-6  40-29  -20     0-87 

8  22  28  12-1  -23-27 

-7     356 

9  22  21   85  -2310 

-7  26-66 

10  22  13  41-9  -22-93 

-7  49-59  -23  2877 

11  22     5  52-3  -2278 

-8  12-37 

12  21  57  39-9  -22'61 

-8  34-98 

13  21  49     4-9  -22-42 

-8  57-40  -26  52-21 

14  21  40     7'5  -23-19 

-9  19-59 

15  21  30  479  -21-97 


17.]  INTERPOLATION   TO  THIRDS  33 


To  make  the  process  in  the  example  clear,  the  computed 
differences,  etc.,  are  printed  in  heavier  type  than  the  interpo- 
lated ones. 

It  is  also  to  be  remarked  that  the  sum  of  the  three  consecutive 
values  of  <T,  formed  of  one  computed  value  and  the  interpolated 
values  next  above  and  below  it,  should  be  equal  to  the  difference 
between  the  corresponding  computed  first  differences.  For 

instance, 

"-93  =  7'  49"'59-6'  40"'29. 


But  in  the  first  computation  this  condition  will  seldom  be 
exactly  fulfilled,  owing  to  the  errors  arising  from  omitted 
decimals  and  other  sources.  If  the  given  quantities  are  accurate, 
the  errors  should  never  exceed  half  a  unit  of  the  last  decimal  in 
the  given  quantities,  or  five  units  in  the  additional  decimal 
added  on  in  dividing. 

To  correct  these  little  imperfections  after  the  interpolation  of 
the  second  differences,  but  before  that  of  the  first  differences,  the 
sum  of  the  last  two  figures  in  each  triplet  of  second  differences 
should  be  formed,  and  if  it  does  not  agree  with  the  difference  of 
the  first  differences,  the  last  figures  of  the  second  difference 
should  each  be  slightly  altered,  to  make  the  sum  exact. 

The  first  difference  can  then  be  formed  by  addition. 

In  the  same  way,  the  sum  of  three  consecutive  first  differences 
should  be  equal  to  the  difference  between  the  given  quantities. 
If,  as  is  generally  the  case,  this  condition  is  not  exactly  fulfilled, 
the  differences  should  be  altered  accordingly.  This  alteration 
may,  however,  be  made  mentally  while  adding  to  form  the 
required  interpolated  functions. 

17.  Interpolation  to  fourths. 

This  may  be  effected  by  two  successive  interpolations  to  halves. 
The  processes  may  be  combined  thus  : 
Let  us  put 

^1J    ^2>    ^3J    ^4' 

the  four  first  differences  of  the  interpolated  series,  so  that 


N.S.A. 


34  DIFFERENCES,   INTERPOLATION,   DEVELOPMENT    [§17. 

Then,  by  (14),  we  have 

1    ,      1  A' 


128       2 
3 


2          128        2 

From  Bessel's  formula,  by  putting   t  —  ^,  -J,  -J  in  succession 
we  find 

J^  A£+A?    J_  A,,,  __  11_ 
62     dl~16       2          64  Ai     1024  "     2 

i   /.    n 


«    .  _ 

0)4     °3~16       2 
In  practice  we  first  compute 


1024 


and  then 


51  =  (l)-(2) 


=  (3)  -(4), 


.(18) 


18.  Interpolation  to  fifths. 

Let  us  next  investigate  the  formulae  when  every  fifth  quantity 
is  given  and  the  intermediate  ones  are  to  be  found  by  inter- 
polation. By  putting  t  =  %  in  the  Besselian  formula,  we  shall 
have  the  value  of  the  interpolated  function  second  following  one 
of  the  given  ones,  and  by  putting  £  =  -§•  that  third  following. 
The  difference  will  be  the  middle  interpolated  first  difference  of 
the  interpolated  series. 

Putting  £  =  -f  in  (10),  we  have 

,2          2.3AQ  +  A1,    2.3.1   A,,,  ,  2.3.7.8 


2.3.7.8.1 


Z 


22.3.4.5.55 


18.]  INTERPOLATION  TO  FIFTHS  35 

Putting  t  =  f  ,  we  have 

3      2.3  A;'+A;'    2.3.1       2.3.7.8  A 

™f-^o+5Ai     2.55       2          2273^3ZH  +  2.3.4.54 

2.3.7.8.1 


22.3.4.5.55 

The  difference  of  these  expressions,  being  reduced,  gives 


' 


25V"*     125 L 

The  term  in  Av  will  not  produce  any  effect  unless  the  fifth 
differences  are  considerable,  and  then  we  may  nearly  always,  in 
practice,  put  ^  instead  of  ^^- 

The  interpolated  second  differences  opposite  the  given  func- 
tions are  most  readily  obtained  by  Stirling's  formula  (9). 
Putting  t  =  ^,  we  have  the  following  value  of  the  interpolated 
first  differences  immediately  following  a  given  value  of  the 
function : 

1  24 


24 


6 .  5 .  20 .  25 


Again,  putting  £  =  —  -J-,  and  changing  the  signs,  we  find  for  the 
first  difference  next  preceding  a  given  function 


24 


24 


^6.5.20.25" 

The  difference  of  these  quantities  gives  the  required  second 
difference,  which  we  find  to  be 


36  DIFFEKENCES,   INTERPOLATION,   DEVELOPMENT     [§  18. 

As  an  example  and  exercise  we  show  the  interpolation   of 
logarithms  when  every  fifth  logarithm  is  given : 


Number.  Logarithm. 

1000  3-000  000  0 

1005  3-0021661 

1006  -002  598  0 

1007  '003  029  5 

1008  -003  460  6 

1009  -003  891  2 

1010  3-0043214 

1011  -004  751  2 

1012  -005  180  5 

1013  -005  609  4 

1014  -006  037  9 

1015  3-006  466  0 
1020  3-008  600  2 


d' 

d" 

A' 

+  21661 

-432 

4319-2 

-431 

4314-9 

-4-30 

4310-6 

-  4-30 

+  21553 

4306-3 

-4-29 

4302-0 

-4-28 

4297-7 

-4-27 

4293-5 

-4-26 

4289-2 

-4-23 

+  21446 

4285-0 

-4-20 

4280-8 

-416 

-108 


-107 


-104 


+  21342 


19.  Numerical  differentiation  and  integration. 

When  the  numerical  values  of  a  function  for  a  series  of 
equidistant  values  of  its  argument  are  given,  both  the  differential 
coefficients  of  the  function,  and  its  integral  between  any  two 
values  of  the  argument,  may  be  found. 

From  the  successive  differences  of  the  values  of  a  quantity  we 
may  find  not  only  intermediate  values,  but  the  derivatives  as  to 
the  argument.  Taking  as  a  unit  of  the  argument  t  the  intervals 
of  the  series,  we  find,  by  expressing  (9)  in  powers  of  t  and 
differentiating, 


20.] 


NUMERICAL   DIFFERENTIATION 

//  2 


37 


Dtut  =  A;-A;''  +      A0v-... 


.(20) 


etc.  etc. 

If  the  interval  of  the  argument  is  k  units  the  nih  derivative 
thus  obtained  will  be  kn  times  too  large,  and  we  shall  have  for 
the  true  values 


20.    Development  in  powers  of  the  time. 

The  preceding  formulae  enable  us  to  develop  a  quantity  in 
powers  of  the  time  when  we  have  given  a  sufficient  number  of 
special  values  of  the  quantity  for  equidistant  epochs.  As  an 
example  of  the  method  we  shall  take  the  values  of  a  certain 
quantity  X,  which  enters  into  the  theory  of  precession,  and  for 
which  we  shall  hereafter  derive  the  values  shown  in  the  follow- 
ing table.  The  table  shows  also  the  successive  orders  of 
differences,  so  far  as  they  are  required. 

Epoch.  X  A'  A"  A'" 

1850  0"-000 


(22) 


1900 

6  114 

-r-193 

4-921 

-1 

1950 

11  -035 

-1  -192 

3-729 

0 

2000 

14  -764 

-1  -192 

2  -537 

0 

2050 

17  -301 

-1  -192 

1  -345 

2100 

18  -646 

'  ••'•*•"£ 

38  DIFFERENCES,  INTERPOLATION,  DEVELOPMENT     [§  20. 

In  this  problem  it  is  best  to  first  take  a  date  near  the  middle 
of  the  series  as  the  initial  one.  We  shall  therefore  count  t  from 
1950,  using  50  years  as  the  unit.  We  then  have 


A;'  =  -l"'l  92, 
Ao'=-  0"'0005. 

By  substituting  these  values  in  Stirling's  formula,  the  develop- 
ment becomes 

(23) 


For  further  use  we  transfer  the  epoch  from  1950  to  1850,  and 
express  the  time  in  terms  of  the  century  as  the  unit.  Putting 
t  for  the  time  thus  expressed,  we  have 


Substituting  these  values  of  t  and  t2,  we  find 

(24) 


In  applying  this  process  it  must  be  noted  that,  if  we  make  the 
development  conform  exactly  to  the  special  numerical  values, 
the  imperfections  of  the  decimals  may  result  in  adding  fictitious 
terms  to  the  series.  We  must  therefore  drop  all  coefficients  of 
powers  of  t  from  the  point  where  the  differences  cease  to  be 
regular.  In  the  present  case  it  is  evident  that  the  second  differ- 
ences may  be  taken  as  constant  ;  we  therefore  stop  at  the  term 
in£2. 

After  obtaining  the  development  in  this  way  it  is  advisable  to 
compare  its  results  with  the  special  values  of  the  quantity,  and 
correct  the  coefficients  so  as  to  secure  the  best  representation  of  the 
given  quantities.  In  the  present  case  we  shall  find  that  the  six 
special  values  of  X  are  all  represented  exactly  by  the  expression 
(24)  except  the  first.  The  rigorous  value  is  0  at  the  epoch  1850. 
If  we  drop  the  first  term  from  (24),  all  the  other  values  will  be  in 
error  by  0"*001.  We  can  lessen  this  difference  by  a  slight  change 
in  the  coefficients  of  T  and  T2,  adding 

AX  =  0"'0010r-  0"-0002T2. 


§  20.]          DEVELOPMENT   IN  POWERS  OF  THE  TIME  39 

Applying  this  correction  to  (24),  the  definitive  value  of  X  will 
then  be 

X  =  13"'4190T-  2"-3842T2. 

The  special  values  for  the  six  epochs,  and  their  deviations 
from  the  computed  values  (22),  are  as  follows  : 


1850 

0"-0000 

Dev.  =  0 

1900 

6  -1134 

-6 

1950 

11  '0348 

-2 

2000 

14  7640 

0 

2050 

17  -3012 

+  2 

2100 

18  -6462 

+  2 

The  integral  of  the  function  u  may  be  found  by  integrating 
its  value  developed  in  powers  of  the  time.  But  this  method 
is  limited  in  its  application.  The  integrals  through  long  periods 
of  time  are  found  by  the  process  of  mechanical  integration, 
which,  not  being  necessary  in  spherical  astronomy,  is  not 
developed  in  the  present  work. 


NOTES  AND  REFERENCES. 

The  mathematical  theory  of  interpolation  is  developed  by  Gauss  in  a 
memoir  on  that  subject ;  but  from  a  point  of  view  wholly  different  from 
that  of  the  practical  computer. 

OPPOLZER  in  his  Lehrbuch  zur  Bahnbestimmung,  vol.  ii.,  Introductory 
Chapter,  develops  the  general  formulae  with  great  fulness,  having  especially 
in  view  the  process  of  mechanical  integration. 

RICE,  HERBERT  L.,  The  Theory  and  Practice  of  Interpolation,  Lynn,  Mass., 
Nichols  Bros.,  1899,  is  a  very  copious  and  extended  exposition  of  the  various 
applications  of  the  method,  which  can  be  recommended  to  the  computer  who 
has  much  of  this  work  to  do. 


CHAPTEK  III. 

THE  METHOD   OF  LEAST   SQUARES. 
Section  I.    Mean  Values  of  Quantities. 

21.  The  "method  of  least  squares"  is  a  subject  which  requires 
a  volume  for  its  full  treatment.     But  the  most  essential  prin- 
ciples involved  in  it,  and  the  simplest  of  the  processes  which  are 
applied  in  much  every-day  astronomical  work,  can  be  set  forth 
in  a  smaller  compass. 

The  method  has  its  origin  in  the  fact  that  when  we  aim  at  the 
highest  precision  in  astronomical  measurement,  we  find  the 
results  of  our  measures  to  be  affected  by  small  errors  due  to  a 
multiplicity  of  unavoidable  causes.  Some  of  these  are  in  the 
nature  of  accidents ;  of  others  the  causes  are  known  in  a  general 
way,  but  cannot  be  obviated  or  determined  in  detail.  The  result 
is  that  a  perfect  agreement  between  two  observations  is  never 
to  be  expected.  The  combination  of  discordant  measures  so  as 
to  derive  the  most  likely  result  thus  becomes  an  important  part 
of  the  astronomer's  work. 

22.  Distinction  of  systematic  and  fortuitous  errors. 

The  errors  in  astronomical  measurement  are  divided  into  two 
classes,  one  called  systematic,  the  other  accidental  or  fortuitous, 
according  to  the  nature  of  their  causes. 

Systematic  errors  are  those  arising  from  causes  which  continue 
their  action  through  a  series  of  observations,  or  are  in  any  way 
governed  by  a  determinable  law.  Examples  of  such  causes  are : 
Changes  of  temperature  which  may  cause  an  instrument  to  give 


^  22.]  SYSTEMATIC   AND  FORTUITOUS   ERRORS  41 

a  different  result  in  summer  and  in  winter,  or  during  the  day  or 
in  the  night ;  varying  conditions  of  the  atmosphere,  resulting  in 
its  refracting  light  differently  on  one  night  from  what  it  does 
on  another;  habits  of  the  observer  leading  him  to  make  an 
error  of  the  same  general  nature  in  a  whole  series  of  obser- 
vations; imperfections  in  the  construction  of  an  instrument 
leading  to  its  results  being  always  erroneous  in  a  more  or  less 
regular  way. 

A  systematic  error  of  which  the  amount  is  always  the  same 
is  called  constant.  This  term  is  also  applied  to  the  mean  value 
of  any  systematic  error  in  a  series  of  observations.  An  example 
of  a  constant  error  is  offered  by  a  scale  of  millimetres  or  angles 
being  too  long  or  too  short.  It  is  evident  that,  in  every  such 
case,  all  the  measures  made  with  the  scale  will  be  too  small  or 
too  large  by  a  corresponding  amount.  If  the  scale  is  correct  at 
a  certain  standard  temperature,  and  the  observer  uses  it  at 
another  temperature,  always  higher  or  lower  than  the  standard 
one,  the  general  mean  of  the  systematic  errors  will  be  those 
corresponding  to  the  mean  of  the  actual  temperatures. 

The  general,  though  not  the  universal,  rule  is  that  systematic 
errors  admit  of  investigation  and  determination,  so  that  we  may 
with  more  or  less  certainty  determine  the  proper  corrections  to 
be  applied  in  order  to  annul  their  effect. 

Accidental  or  fortuitous  errors  are  those  of  which  the  causes 
are  so  variable  and  transient  that  the  resulting  errors  elude 
investigation.  For  example,  if  an  observer  seeks  to  bisect  the 
segment  of  a  line  by  his  eye  and  by  estimation,  there  must  be  a 
range  of  accidental  error,  at  least  equal  to  the  smallest  space 
perceptible  to  the  senses.  The  undulations  of  the  air,  which 
never  entirely  cease,  cause  the  image  of  a  star,  as  seen  in  a 
telescope,  to  be  continually  affected  by  a  small  and  irregular 
motion,  or  change  of  form.  An  error  which  cannot  be  estimated 
in  advance  will  therefore  be  made  by  the  observer  when  he 
attempts  to  bisect  the  image  with  a  spider  line. 

The  general  rule,  in  astronomy  at  least,  is  that  such  accidental 
errors  are  the  result  of  the  separate  action  of  a  multiplicity  of 
causes,  too  variable  and  complex  to  be  individually  determined 


42  THE   METHOD  OF  LEAST   SQUARES  [§  22. 

or  even  defined.  The  theory  of  such  errors  and  the  reduction  of 
their  injurious  effect  to  a  minimum  form  the  basis  of  the  subject 
commonly  known  as  the  Method  of  Least  Squares. 

This  method  properly  applies  only  to  accidental  errors.  The 
best  results  we  can  derive  by  it  will  still  be  affected  by  the 
constant  or  average  effect  of  all  the  systematic  errors  to  which 
the  observations  are  liable.  These  we  must  determine  as  best 
we  can  in  each  case,  and  regard  the  uncertainties  of  the  deter- 
mination as  belonging  to  the  class  of  accidental  sources  of  error. 

23.  The  arithmetical  mean  and  the  sum  of  the  squares  of  residuals. 

We  begin  with  the  simplest  case,  which  is  that  of  the  repeated 
measurement  of  a  constant  quantity,  which  we  may  call  x. 

Let  the  individual  results  of  n  measurements  be  that  this 
quantity  is  found  to  have  the  values 

*^l>  *^2>  *^3>  *  *  *  ^n' 

We  may  express  these  results  by  saying  that  they  lead  to  the 
discordant  equations 

Ji  — HZj, 

X  =  Xt 


or 


.and  the  question  is  how,  from  all  these  equations,  we  are  to 
conclude  upon  the  best  value  to  adopt  for  x. 

Let  us  first  regard  x  as  an  indeterminate  quantity,  to  which  we 
may,  as  an  hypothesis,  assign  any  value  at  pleasure.  We  may 
form  the  deviations  of  any  such  value  from  the  observed  values. 

Putting 

for  these  deviations,  we  have 


§  24.]  THE  AEITHMETICAL   MEAN  43 

Now  let  us  form  the  sum  of  the  squares  of  these  residuals, 
which  we  call  Q. 


This  sum  is  a  quadratic  function  of  x;  and  the  fundamental 
principle  adopted  is  that  the  most  likely  or  best  value  of  x  to 
be  chosen  is  that  which  makes  the  sum  Q,  of  the  squares  of  the 
residual  differences  the  least  possible.  To  find  this  value  we 
differentiate  £2  as  to  x,  and  equate  the  derivative  to  zero.  We 
thus  find  /7O 


which,  being  equated  to  zero,  gives 


Ti 

Hence,  on  the  principle  in  question,  the  most  likely  value  of  the 
quantity  measured  is  the  mean  result  of  the  individual  measures. 

24.    The  probable  error. 

Since  every  astronomical  result  is  liable  to  error,  we  need 
some  way  of  expressing  the  amount  of  the  liability.  We  may 
do  this  by  assigning  a  quantity  e  such  that  we  suppose  it  a 
certain  definite  chance  whether  the  observation  is  in  error  by 
an  amount  greater  or  less  than  e.  It  is  common  to  regard  the 
chance  in  question  as  an  even  one  ;  the  value  of  e  is  then  called 
the  probable  error. 

The  judgment  that  a  numerical  value  xv  assigned  to  x,  is 
affected  by  the  probable  error  e,  that  is  to  say,  that  the  true  x 
probably  differs  from  xl  by  the  quantity  e,  is  expressed  in  the  form 

#  =  xi  ±  6. 

This  means  that,  out  of  four  chances,  there  are  two  that  x  is 
contained  between  the  limits  xl  —  e  and  a31  +  ^>  and  two  that  its 
true  value  lies  without  these  limits.  Let  us  lay  down  the  value 
of  x  graphically  on  an  axis  of  abscissae  and  measure  off  the 
value  of  the  probable  error  on  each  side,  thus 

i    i    i 

-e    0     +e 


44  THE   METHOD  OF  LEAST   SQUARES  [§  24, 

The  point  0  here  marks  the  adopted  value  of  x,  while  +e  and 
—  e  are  laid  off'  on  each  side. 
Then  the  four  chances  that  the  true  value  of  x  lies 

To  the  left  of  -6     \ 

Between  —  e  and  0 

I  /  A  \ 

Between  0  and  +e  ! 
To  the  right  of  +e  } 
are  all  equal. 

The  following  problems  are  fundamental  in  the  whole  theory  : 

PROBLEM  I.  The  probable  error  of  x  being  +  e,  to  find  the 
probable  error  of  mx,  m  being  a  constant. 

If  x  is  contained  between  the  limits  x  +  e  and  x  —  e,  mx  will  be 
contained  between  the  limits  mx  —  me  and  mx+me. 

There  is,  therefore,  the  same  chance  that  mx  will  be  contained 
between  these  limits  as  there  is  that  the  true  x  will  be  between 
x  +  e  and  x  —  e. 

Hence  probable  error  of  mx  =  +me  ....................  (o) 

It  may  be  noted  that  this  theorem  is  true  of  values  of  m 
either  greater  or  less  than  unity. 

Definition.  Independent  quantities  are  such  that  a  change  in 
the  adopted  value  of  one  will  not  affect  our  judgment  of  the 
value  of  the  other. 

PROBLEM  II.  The  probable  errors  of  several  independent 
quantities  of  ivhich  the  given  values  are  qv  q2,  ...  qn  being 
ev  ez,  ...  en  :—to  find  the  probable  error  of  their  sum. 

Let  us  put 

qt  the  true,  but  unknown,  value  of  qt...  ,     (*  =  1,  2,  ...  n) 
Q  =  q1  +  q2  +  .  •  .  +  qn>  the  sum  as  given, 
Q'  =  q(  +  q'2  +  .  .  .  +  qn,  the  true  unknown  sum. 
We  may  then  write  the  equations 


which  will  mean  that,  in  each  equation,  it  is  an  even  chance 
whether  the  second  member  is,  in  absolute  magnitude,  greater 
or  less  than  the  value  e,  which  is  assigned  to  it. 


.$  24.]  THE   PEOBABLE   ERROR  45 


Take  the  sum  of  all  these  equations, 

' 


Q 
and  square  it 


, 

'-Q=  ±e1±e,±e3±...±e 


±  e&  ±...±  en-&n. 

Since  the  quantities  e  are  equally  likely  to  have  positive  or 
negative  values,  the  sum  of  the  terms  in  the  last  line  is  as  likely 
to  be  positive  as  negative.     The  probable  average  value  of  this 
sum  is  therefore  zero,  and  the  most  probable  value  of  the  second 
term  of  the  equation  becomes  e*  4-  e}  +  .  .  .  +  en*.      We  therefore 
obtain  as  the  most  likely  value  of  Q'  —  Q, 

3=V-Q  =  JW  +  ef  +  e,s+...+e*),  ..............  (6) 

E  being  put  for  the  probable  error  of  the  sum  Q.     We  therefore 
have  the  theorem  : 

The  probable  error  of  the  sum  of  any  number  of  independent 
quantities  is  equal  to  the  square  root  of  the  sum  of  the  squares 
of  the  probable  errors  of  the  individual  quantities. 

We  note  that  the  seeming  difference  between  the  conclusions 
in  the  two  cases  of  Problems  I.  and  II.  arises  from  the  premise 
of  the  second  case  that  the  quantities  are  independent. 

PROBLEM  III.  To  find  the  probable  error  of  a  linear  function 
of  several  independent  quantities  in  terms  of  the  probable  errors 
of  the  separate  quantities. 

Let  the  quantities  with  their  probable  errors  be 

*1±«1,     ^2  +  ^2,     ^3  +  ^3,      •••, 

and  let  the  linear  function  be 

X  =  ax1  +  bx2  +  cx3+  — 
By  Problem  I.  we  have 

Probable  error  oi.'axl=  +  06^ 


cx3=±ce3, 


and  then,  by  Problem  II., 

Probable  error  of  X  =  J(a*e*  + We *  +  <*e* +...), (7) 

which  is  the  required  result. 


46  THE   METHOD  OF  LEAST   SQUARES  [§  24. 

PROBLEM  IV.  To  find  the  probable  error  of  the  arithmetical 
mean  of  several  independent  quantities  each  having  the  same 
probable  error. 

The  arithmetical  mean  of  m  quantities  x^  x2  ...xm  being  the 
linear  function 

^1  -|_  ^2    i    ^3   i  ,Xm 

m    m    m  m' 

this  case  becomes  identical  with  that  of  Problem  III.  by  putting 

a  —  b  —c  =  ...=—, 
m 


Hence,  by  substitution  in  (7), 

//vvi  />«  /> 

Probable  error  =  \l  — ?  =  — ^=.  ...  (8) 

*  m*     +Jm 

Hence:  The  probable  error  of  the  mean  result  of  several 
observations  of  the  same  quantity  is  inversely  as  the  square 
root  of  their  number. 

25.  Weighted  means. 

The  arithmetical  mean  of  several  results  can  be  the  most 
likely  value  only  in  the  case  when  we  have  no  reason  to  prefer 
any  one  of  the  results  to  any  other.  But  if  some  of  the  results 
have  a  smaller  probable  error  than  others,  it  is  evident  that 
we  should  esteem  them  of  greater  reliability  in  reaching  a 
conclusion. 

To  find  how  we  must  modify  the  principle  of  least  squares 
in  this  case,  suppose  that  the  given  results 

«^1 )    ^2 '  * '  *  ^n ' 

instead  of  being  individual  results  of  separate  observations,  are 
each  the  mean  of  several  observations  of  one  and  the  same 
quantity  throughout,  namely : 

xlt  the  mean  result  of  ml  obs.; 

**' "  "  "  "        ^"        " 


§26.]  WEIGHTED   MEANS  47 

The  mean  we  should  adopt  is  that  of  the  original  results. 
To  form  their  sum  we  note  that,  since  xl  is  the  mean  of  mx 
quantities,  the  sum  of  these  quantities  must  have  been  m^. 
Hence  the  sum  of  all  the  products  m^,  m2x2,  etc.,  will  be  the 
sum  of  the  original  results,  while  the  number  of  the  latter  is 
the  sum  of  the  m's.  Hence  the  required  mean  is 

. 


In  this  expression  the  factors  m  are  termed  weights  and  the 
result  is  called  a  weighted  mean,  or  a  mean  by  weights  of  the 
quantities  x±  .  .  .  xn 


0        xl  x2   xB...Xi    xn      X 

This  conception  of  a  weighted  mean  has  a  mechanical 
analogue.  If  we  lay  off  on  an  axis  OX  the  values  of  xlt  xz,  etc., 
the  arithmetical  mean  of  all  the  measures  from  0  to  X  corresponds 
to  the  distance  from  0  to  the  centre  of  gravity  of  the  points 
xlt  x.2,  ... ,  when  all  are  assigned  equal  weights.  If  we  imagine 
the  points  to  have  different  weights,  the  position  of  the  centre 
of  gravity  is  that  of  the  weighted  mean  of  the  quantities. 

It  is  evident  that  if  all  the  numbers  m  are  multiplied  by  any 
common  factor,  the  resulting  value  of  x  will  not  thereby  be 
changed.  Hence  we  may  take  for  the  weights  any  system  of 
numbers  proportional  to  the  respective  numbers  of  observations 
from  which  each  separate  result  is  derived.  In  other  words,  we 
may  find  the  weights  by  multiplying  or  dividing  the  numbers  of 
observations  by  any  common  factor  or  divisor.  We  represent 
weights,  in  a  general  way,  by  the  symbols 

wlt  w2,...wn. 

26.  Relation  of  probable  errors  to  weights. 

In  (9)  the  weights  m  are  the  respective  numbers  of  observa- 
tions on  which  each  x  depends.  But,  suppose  that,  instead  of 
the  number  of  observations  being  given,  we  have  given  the 
probable  error  of  each  measure  or  series  of  measures.  It  is 
evident  that  the  final  result  to  be  derived  should  depend  on 


48  THE   METHOD  OF   LEAST   SQUARES  [§  26. 

these  probable  errors,  irrespective  of  the  number  of  observations 
which  enter  into  each  result. 

It  has  been  shown  (Prob.  IV.)  that  the  probable  error  of 
any  number  of  observations,  each  having  the  same  individual 
probable  error,  is  inversely  proportional  to  the  square  root  of 
the  number.  If  e0  be  the  probable  error  of  a  single  observa- 
tion, that  of  the  mean  of  m  observations  will,  therefore,  be 


Hence  m  =     .  .  ...(10) 

ez 

This  result  is  the  converse  of  (8),  and  shows  that  the  number 
of  observations  necessary  to  give  a  result  with  an  assigned 
probable  error  is  inversely  as  the  square  of  that  error. 

It  follows  that  if  we  have  given  the  results 


with  the  respective  probable  errors 

we  may  choose  at  pleasure  a  quantity  e0  as  the  probable  error 
of  a  fictitious  standard  observation,  to  which  we  assign  the 
weight  1,  and  then  find  the  series  of  numbers 

Wl  =  Tt>       U''2==  ~^~2>  "•»       Wn~^TlL>    (•**) 


which  will  be  the  respective  numbers  of  fictitious  observations 
the  means  of  whose  results  will  have  the  given  probable  errors. 
There  is  no  need  that  the  numbers  w  shall  be  integers ;  nor  is 
there  commonly  any  practical  advantage  in  writing  their  values 
with  more  than  a  single  significant  digit,  or,  at  most,  a  pair  of 
digits  when  the  first  digit  is  1. 

As  a  concrete  example,  suppose  the  seconds  of  mean  declination 
of  a  star  at  a  certain  epoch,  as  determined  at  several  observatories, 
with  their  probable  errors,  to  be 

<5  =  3"-l;  3"-7;  2"-9,  3"'2,  3"7, 

2;  0"-25;  0*18;  0"'13;  0"'40. 


§  27.]    RELATION   OF  PROBABLE   ERRORS  TO  WEIGHTS  49 

As  a  convenient  round  number  we  choose  e0  =  0"-50  as  the 
probable  error  of  a  fictitious  standard  observation.  We  then 
have  the  following  computation  from  (11)  and  (9): 

Wt 


1. 

1 

0"'048 

5 

15"-5 

2 

0  -062 

4 

14-8 

3 

0-032 

8 

23  -2 

4 

0  -017 

15 

48  -0 

5 

0-160 

2 

7  -4 

34          108"-9 
Weighted  mean  :  108"'9  -r-  34  =  3"-20. 

PROBLEM  V.     To  find  the  probable  error  of  a  weighted  mean. 

Since  the  weighted  mean  may  be  regarded  as  that  of  a  certain 
number  of  standard  observations,  its  probable  error  is  given  by 
(8).     The  number  in  question  being 

Wl  +  w2+  ...+Wn=W,  .....................  (12) 

the  probable  error  e  of  x  is 


or  the  quotient  from  dividing  the  probable  error  of  a  fictitious 
observation  of  weight  1  by  the  square  root  of  the  sum  of  the 
weights. 

The  same  result  may  be  reached  in  an  elegant  way  by  Problem 
III.  A  weighted  mean  is  a  linear  function  of  the  quantities 
whose  mean  is  taken,  of  which  the  coefficients  are 

Wl  7,        WZ 

a==W>     b=W' 

Substituting  in  (7)  these  values  of  a,  6,  etc.,  and  putting  for 
•ej2,  e},  etc.,  their  values  from  (11) 


the  probable  error  reduces  to  the  expression  (13). 

27.  Modification  of  the  principle  of  least  squares  when  the  weights 
are  different. 

It  is  evident  that,  when  we  take  a  weighted  mean  of  several 
quantities,  the  sum  of  the  squares  of  the  residuals  can  no  longer 
N.S.  A.  D 


50  THE   METHOD   OF  LEAST  SQUARES  [§  27. 

be  a  minimum,  because  this  is  the  case  with  the  unweighted 
mean.  To  find  the  corresponding  function  which  is  a  minimum, 
let  the  quantities  whose  weighted  mean  is  taken  be,  as  before, 

*^l  >    ^2  '    *^3  '    *  '  *   ^n  > 

and  their  probable  errors 

^1  >    ^2  '    ^3  '   '  '  '    ^w 

Their  respective  weights  will  then  be 


If  we  multiply  the  equations  (1)  or  (2)  by  the  respective  factors 

Jwi  =  %,    x/m,  =  ^,etc  ...................  (15) 

el  e2 

it  follows  from  (5)  that  the  probable  errors  of  the  products 


ii  22   .  .  . 

will  all  be  equal  to  e0.  By  applying  this  multiplication  to 
equation  (2)  the  second  members  will  become  Jw^r^  Jw2r2,  ...  , 
and  it  is  the  sum  of  the  squares  of  these  products,  that  is,  the 
function  Wp*  +  w^f  +  .  .  .  +  ^nrn  =  Q,  ..................  (16) 

which  should  be  a  minimum.  If  we  substitute  for  rv  r9)  ... 
their  values  (2),  differentiate  as  to  x,  and  equate  to  0,  we  shall 
have  Wi(x  _  Xi)  +  w^x  _  x^  +  .  .  .  =  Q, 

which  will  give  for  x  the  weighted  mean  of  xlt  x2,  etc. 

28.  Adjustment  of  quantities. 

It  sometimes  happens  that  we  know  in  advance  some  relation 
which  a  system  of  measured  quantities  should  satisfy.  For 
example,  if  at  a  point  0  on  a  horizontal  plane,  surrounded  by 
points  A,  B,  C,  D,  ...  K,  lying  on  the  plane  in  various  directions, 
we  measure  the  consecutive  angles 

AOB,  BOG,  COD,  ...  KOA, 

the  last  angle  carrying  us  round  to  the  starting  direction,  then 
we  know  that  the  sum  of  these  angles  should  be  360°.  If  the 
measures  were  free  from  error,  then,  on  adding  them  together, 
the  sum  would  be  exactly  360°.  If  the  actual  sum  is  different 
from  this  by  a  quantity  A,  we  know  that  A  is  the  algebraic  sum 


§  28.]  ADJUSTMENT  OF  QUANTITIES  51 

of  all  the  errors  of  the  measures.  The  problem  then  is  to  find 
the  most  likely  system  of  corrections,  which,  being  applied  to 
the  individual  measures,  will  reduce  their  sum  to  360°.  In  doing 
this  we  are  adjusting  the  measures  so  as  to  fulfil  the  required 
conditions,  for  which  reason  the  process  is  called  adjustment. 
If  the  measures  are  all  of  one  weight,  the  process  of  adjustment 

is  this  :  —  Putting 

av  az,  aB,  ...  an, 

the  measured  values  of  the  successive  angles,  then 


will  be  the  sum  of  the  errors.     The  most  likely  adjustment  will 
then  be  to  divide  the  errors  equally  among  all  the  angles.     If 

we  put  A 

8  =  —  > 

n 

the  concluded  values  of  the  separate  angles  will  be 

±r  o 

04  —  S,  a2  —  3,  .  .  .  an  —  S, 

the  sum  of  which  fulfils  the  required  condition  of  being  360°. 
If  the  weights  are  unequal,  let  us  put  for  the  weights  of  the 

n  measures  a     a     a          a 

u-p  a2,  c£3,  ...   an 

the  symbols  w^  w^  ^  _  Wnt 

and  let  the  respective  corrections  be 

hv  h2,  h3,...  hn, 
The  sum  of  these  quantities  must  satisfy  the  condition 

h1  +  h2  +  hB+...+hn=  -A.  ..................  (17) 

while,  in  accordance  with  the  general  principle,  the  function 

Q  =  wji*  -f  wji/  +...+  wnhnz 
must  be  the  least  possible.     Hence  the  equation 

wji^dhi  +  wji2dh2  +...+  wnhndhn  =  0  ............  (18) 

must  be  satisfied  for  all  values  of  the  dh'a  which  satisfy  the 
equation  given  by  the  differentiation  of  (17),  namely 

dhl  +  dh2  +  dh3+...  +  dhn  =  0  ..................  (19) 

These  conditions  must  hold  true  for  every  admissible  infini- 
tesimal  change   in   the   values  of  .the  h's.     To  find  the   values 


52  THE   METHOD   OF  LEAST   SQUARES  [§28. 

which  satisfy  the  conditions,  we  multiply  (19)  by  an  indeter- 
minate factor  X  and  subtract  the  product  from  (18),  thus 
obtaining 

(wjt^  —  X)^  +  (w2h2  —  X)  dh.2  -f  .  .  .  +  (wnhn  —  X)  dhn  =  0. 
In  order  that  this  equation  may  be  satisfied  for  all  values 
dhy  etc.,  we  must  have 

wJh  —  ^  =  ()>  W2^2  —  ^  =  0,  etc.  , 
or  wji^  =  w2A2  =  wBh3  .  .  .  =  wnhn  =  X. 


Hence  h,  =  — 


(20) 


•h 

fin  =  — 

wn 

Equating   the   sum   of   these   quantities   to   —A  gives  us  an 
equation  for  determining  X  ; 

-A 


W,  '" 


.(21) 


'•%  ^^9  ^^71 

which    substituted    in   (20)   gives   the   values   of  the   adjusted 
corrections  h. 

It  is  interesting  and  instructive  to  note  the  form  of  this  result 
when  one  of  the  measures  has  the  weight  0,  or,  in  other  words, 
is  regarded  as  entirely  worthless.  Let  this  measure  be  the  first 
so  that  w1  =  0. 

Then  (21)  will  give  X  =  0 

and  (20)  will  give          h2  =  h3  =  . . .  =  hn  =  0, 

but  will  leave  \  indeterminate.     But   its   value   is   found   by 
using,  instead  of  (20)  the  expression  (17).     This  gives  at  once 

In  other  words  we  are  obliged,  in  this  special  case,  to  determine 
the  faulty  h  from  the  sum  of  all  the  remaining  measures,  a 
conclusion  evident  in  advance. 


§29.]  OF  PROBABLE   AND  MEAN  ERRORS  5S 


Section  II.    Determination  of  Probable  Errors. 

29.    Of  probable  and  mean  errors. 

The  preceding  theory  assumes  as  one  of  the  data  to  be  given 
a  certain  quantity  called  a  probable  error.  The  definitions  of 
this  and  certain  associated  quantities,  and  the  methods  of  deter- 
mining them,  are  now  to  be  considered.  The  following  is  the 
logical  basis  of  the  subject : 

(1)  In  a  rigorous  sense,  an  error  consists  in  the  deviation  of 
an  observed  value  from  an  absolutely  true  value.     But  the  latter 
quantity  is  never  considered  as  actually  known.     Hence,  what 
we  have  to  take  as  an  error  is  the  deviation  of  an  individual 
value  from  the  best  value  that  we  are  able  to  determine.     In 
certain  cases  the  term  residual  or  residual  difference  is  applied 
to  this  quantity.     But,  with  the  limitations  we  have  expressed, 
the  use  of  the  usual  term  "error"  should  cause  no  misconstruction. 

(2)  The  probable  error  e,  as  we  have  defined  it,  is  determined 
by  the  condition  that  there  is  an  equal  chance  of  an  error,  in 
any  one  case,  being  greater  or  less  than  e  in  absolute  value. 

But  the  reasoning,  as  it  has  been  set  forth,  is  equally  valid  for 
the  case  when,  instead  of  taking  e  for  the  amount  of  that  error 
which  there  is  an  even  chance  of  committing,  we  take  the  value 
of  an  error  having  a  different  probability  from  this.  We  may, 
for  example,  take  an  error  of  which  there  are  three  chances 
to  one  against  committing.  In  this  case,  in  the  language  of  the 
theory  of  probabilities,  the  probability  of  an  error  exceeding  the 
standard  amount  will  be  |.  The  reasoning  would  then  remain 
the  same  throughout,  the  meaning  of  the  term  probable  error 
being  alone  changed. 

(3)  We  must  distinguish  between  a  probable  and  an  actual 
error.     The  actual  errors  are  numerical  values  of  the  residuals 
which  we  actually  find  in  the  case  of  any  system  of  observation. 
Probable  errors  are  errors  of  which  there  is  a  greater  or  less 
probability  of  making.     Hence  actual  errors  may  be  greater  or 
less  than  probable  ones,  but,  in  the  long  run  and  the  general 
average,  they  should  correspond  to  each  other. 


54  THE   METHOD  OF  LEAST  SQUARES  [§  29. 

(4)  The  term  mean  error  may  be  used  in  various  significations 
which  must  be  distinguished.  In  general,  the  mean  of  several 
quantities  is  equal  to  the  quotient  of  their  algebraic  sum  by 
their  number.  If  used  in  this  sense,  the  mean  of  all  the  devia- 
tions of  a  system  of  quantities  from  their  arithmetical  mean 
would  always  be  zero,  and  the  term  would  be  without 
significance. 

Another  signification  of  the  term  is  the  mean  of  the  numerical 
values  of  all  the  errors,  regardless  of  their  algebraic  sign.  This 
is  called  an  arithmetical  mean  error,  or  average  error,  but  is  not 
much  used  in  practice. 

As  commonly  used,  the  term  mean  error  is  that  quantity 
whose  square  is  the  mean  of  the  squares  of  the  errors  :  that  is, 
it  is  the  square  root  of  the  arithmetical  mean  of  the  squares  of 
all  the  errors. 

It  is  easily  shown  that  this  mean  is  always  greater  than  the 
arithmetical  mean  of  the  errors,  except  in  the  case  when  all  the 
errors  are  equal. 

For,  let  us  take 

e,  the  mean      1         .         ,   _      . 
h  as  just  defined, 
v,  the  average  ) 

of  the  n  errors 

ev  e2)...  en. 

Let  us  also  take  the  difference  between  each  individual  error 
and  the  average,  and  call  it  c,  so  that  we  have 

el  =  v±cl, 


Now,  take  the  sum  of  the  squares  of  these  equations  : 


The  first  member  is,  by  definition,  Tie2,  and  the  last  term  vanishes, 
because  Zc  =  0.     Hence  y-2 

€2  =  V2  +  ^ 

n 
The  excess  of  e2  over  v2  is  a  positive  quantity,  which  vanishes 


§  29.]  OF  PROBABLE  AND   MEAN  ERRORS  55 

only  in  the  special  case  when  the  e's  are  all  equal  (making  the 
cs  all  zero). 

Here  again  we  must  distinguish  between  a  probable  mean 
error  and  the  actual  mean  error  in  any  given  case.  The  latter 
is  a  numerical  result  actually  found  from  the  residuals;  the 
former  is  defined  as  the  mean  error  which  would  be  found  in 
making  an  infinite  number  of  observations  of  the  same  kind, 

c5 

and,  so  far  as  can  be  determined,  under  the  same  conditions,  as 
those  actually  made. 

The  actual  mean  is  found  only  from  the  special  observations  in 
question,  but,  in  determining  a  probable  mean,  we  may  take  into 
consideration  all  the  data  at  our  disposal  for  its  determination. 

30.    Statistical  distribution  of  errors  in  magnitude. 

The  method  of  dealing  with  fortuitous  errors  rests  upon  the 
law  of  their  statistical  distribution  in  magnitude,  that  is  to 
say,  the  respective  probabilities  of  making  errors  of  different 
magnitudes. 

The  following  are  the  assumed  general  laws  of  distribution, 
from  which,  however,  there  may  be  deviations  in  special  or 
extraordinary  cases.  To  the  latter  the  theory  of  the  subject  does 
not  apply.  In  the  cases  to  which  it  does  apply,  the  principles  are : 

(1)  Positive  and  negative  errors  of  any  given  magnitude  are 
equally  probable. 

It  is  readily  seen  that  there  are  many  kinds  of  investigation 
in  which  this  law  does  not  hold  true.  For  example,  in  weighing, 
impurities  in  the  substance  weighed  will  always  result  in 
making  the  apparent  weight  greater  than  that  of  the  pure  sub- 
stance. In  astronomy,  however,  the  law  is  very  near  the  truth. 

(2)  In  any  class  of  observations,  the  probability  of  an  error 
continually  diminishes  with  its  magnitude,  and  we  can  always 
set  a  limit  beyond  which  the  probability  of  an  error  shall  be 
as  small  as  we  please. 

It  is,  however,  impossible  to  set  a  limit  which  an  error  may 
reach,  but  can  never  exceed.  We  can  only  say  that,  the  larger 
a  possible  error,  the  more  unlikely  it  should  be  to  occur. 

The  preceding  laws  are  commonly  embodied  in  the  following 
formula : — If  we  put  h  for  a  certain  modulus  of  error,  then  the 


56 


THE   METHOD   OF  LEAST  SQUARES 


[§30, 


infinitesimal  probability  that   the    error   shall    lie  between  the 
limits  x  and  x  -f  dx  is  assumed  to  be  given  by  the  formula  : 

(22) 


-= 

/K/7T 

This  formula  may  be  graphically  represented  by  the  curve 
shewn  in  Fig.  2,  in  which,  if  the  abscissa  of  any  point  represents 
the  magnitude  of  an  error,  the  ordinate  at  that  point  is  pro- 
portional to  the  probability  of  the  error.  The  point  P  marks  the 
probable  error  and  M  the  mean  error. 


A  M   P  0  P  M  B 

FIG.  2. 

If  we  have  an  indefinitely  great  number  of  errors  distributed 
in  magnitude  according  to  this  law,  we  may  represent  each  error 
by  measuring  off  its  magnitude  from  the  origin  0  to  the  right 
or  left,  according  as  the  error  is  positive  or  negative,  and  then 
marking  it  by  a  point.  Since  the  ordinate  at  each  point  is 
proportional  to  the  number  of  errors  of  corresponding  magnitude, 
it  follows  that  if  we  scatter  the  points  along  the  ordinate  they 
will  be  equally  distributed  over  the  area  contained  between  the 
curve  and  the  axis  of  abscissas  AB.  The  total  number  of  errors 
between  any  two  limits  will  be  proportional  to  the  area  contained 
between  the  corresponding  ordinates. 

The  probabilities  that  an  error  will  exceed  certain  amounts  are 

The  probable  error  itself  -    O'oOO 
The  mean  error  -    O318 

2  times  the  probable  error    0*177 

3  „         „          „          „         0-043 

4  0-007 


§  31.]         DISTRIBUTION   OF  EEROKS   IN   MAGNITUDE  57 

Thus  the  area  PP  will  be  1/2  and  that  between  the  ordinates 
of  mean  error  MM  will  be  0'318  of  the  entire  area  of  the  curve. 
At  the  points  A  and  B,  corresponding  to  errors  of  about  four 
times  the  probable  error,  the  curve  approaches  so  near  the  axis 
that  only  seven  errors  out  of  one  thousand  should  reach  these 
limits. 

The  preceding  law  of  error  is  considered  the  normal  law,  and 
on  it  the  theory  of  the  subject  is  commonly  based.  But,  although 
it  is  a  law  to  which  the  errors  will  commonly  approximate  when 
the  observations  are  carefully  made,  it  cannot  be  regarded  as 
practically  universal.  Indeed,  in  practice,  the  general  rule  is 
that  large  errors  are  more  common  than  the  normal  law  would 
lead  us  to  infer.  For  example,  an  error  five  times  as  great  as  the 
probable  one  should,  on  the  theory,  occur  only  once  in  1300 
times,  but  practically,  it  will  be  found  to  occur  much  oftener. 
The  theory  is,  however,  adopted  because  of  the  simplicity  and 
elegance  of  the  methods  based  upon  it. 

The  practical  astronomer  has  also  to  recognize  the  occasional,, 
and  perhaps  the  frequent  occurrence  of  errors  which .  seem 
abnormally  large.  Such  an  error  may  be  of  a  magnitude  so  great 
that  no  question  can  arise  as  to  its  retention  or  rejection.  A 
wrong  figure  may  have  been  written  down,  or  a  wrong  graduation 
read  by  the  observer.  But  when  the  magnitude  of  the  error  is- 
such  that  it  cannot  be  regarded  as  morally  impossible,  the 
question  of  dealing  with  it  becomes  one  of  great  difficulty,  to  be 
settled  by  common  sense  and  sound  judgment  rather  than  by 
any  theory.  The  general  rule  is  that,  if  the  magnitude  of  a 
residual  exceeds  the  value  which  we  could  reasonably  suppose 
a  fortuitous  error  to  have  among  a  number  of  observations 
no  greater  than  that  which  we  are  combining,  we  must 
regard  it  as  abnormal,  and  reject  the  result  affected  by  it. 

31.  Method  of  determining  mean  or  probable  errors. 

In  combining  observations  an  important  problem  is  that  of 
inferring  the  probable  error  to  which  any  one  observation  should 
be  regarded  as  liable.  This  may  be  done  in  two  ways : 

(1)  We  may  know  from  experience  that  observations  of  a 
certain  class,  made  at  a  certain  observatory,  or  by  a  certain 


58  THE   METHOD   OF  LEAST  SQUAEES  [§31. 

observer,  are  affected  by  a  probable  error  of  a  certain  amount. 
For  example,  meridian  observations  of  declination  are  affected  by 
a  probable  error  which  may  lie  between  +  0"*2  and  +  0"'5.  When 
made  on  objects  near  the  horizon,  the  p.e.  may  even  exceed  the 
latter  limit.  That  of  good  observations  in  R.A.  commonly  lies 
between  +  0S'018  and  ±0S'035. 

(2)  The  probable  error  of  the  individual  observations  of  a  series 
may  be  determined  by  their  mutual  discordances,  or  the  deviation 
of  each  from  the  mean  of  all.  A  result  thus  reached  will  be 
more  reliable  the  greater  the  number  of  observations.  In 
developing  the  method  of  doing  this  we  begin  with  a  numerical 
example,  illustrating  the  combination  of  observations,  and  the 
determination  and  treatment  of  residuals. 

Twelve  observations  of  the  north  polar  distance  of  Aldebaran, 
made  at  the  Royal  Observatory,  Greenwich,  during  the  year 
1899,  gave  the  results  shown  in  the  first  column  of  the  following 
table,  when  reduced  to  the  beginning  of  the  year.  The  degrees 
and  minutes,  73°  41',  are  omitted,  being  the  same  for  the  whole 
series. 

COMBINATION  OF  GREENWICH  OBSERVATIONS  OF 
ALDEBARAN,  1899. 

Sec.  of  N.P.D.  res.  r2. 


Feb.    9 

36"-55 

-0"-49 

0"'24 

Mar.  23 

38  -04 

+  1  -00 

1  -00 

24 

37  -63 

+  0-59 

0  -35 

Apr.  10 

38  -17 

+  1  -13 

1  -28 

19 

36-87 

-0-17 

0-03 

May    6 

36-29 

-0-75 

0  -56 

June  16 

36-64 

-0-40 

0-16 

July  10 

36  -45 

-0-59 

0  -35 

Nov.  16 

37  -74 

+  0-70 

0-49 

Dec.  11 

36  -11 

-0  -93 

0-86 

13 

37  -18 

+  0-14 

0-02 

14 

36-83 

-0  -21 

0-04 

Sum,     - 

84"-50 

+0"-02 

5"«38 

Mean,     - 

37  -04 

§31.]    DETERMINING  MEAN  OR  PROBABLE  ERRORS     59 

The  units  and  decimals  of  seconds  being  added,  give  the  sum 
84"'50,  and  the  mean  7"'04.  There  is  no  use  in  carrying  the 
division  beyond  the  second  decimal,  which  might,  indeed,  have 
been  omitted  from  all  the  separate  observations  without  detract- 
ing from  the  precision  of  the  result. 

This  mean  being  subtracted  from  each  separate  result  gives 
the  apparent  residual  errors  of  the  latter  found  in  the  third 
column.  In  mathematical  theory  their  algebraic  sum  should 
be  0.  As  a  control  upon  the  accuracy  of  the  mean,  we  form  the 
sum,  and  find  it  to  be  +  0"'02.  This  is  because  the  remainder 
0"'02  has  been  neglected  in  dividing  to  form  the  mean. 

We  next  take  the  square  of  each  residual,  dropping  un- 
necessary decimals,  and  find  the  mean  value  of  all  the  squares. 
But,  in  forming  this  mean,  we  use  a  divisor  less  by  1  than  the 
number  of  observations,  for  a  reason  now  to  be  shewn. 

In  determining  a  probable  error  we  must,  in  effect  at  least, 
express  the  result  as  a  linear  function  of  the  observed  quantities. 
So  we  express  the  residuals  as  linear  functions  of  the  observed 
results.  Let  the  latter,  n  in  number,  be 

^1  j    ^2  '    *^3>  *  *  *  ^n  > 

the  mean  is  the  sum  of  these  divided  by  n.     Subtracting  this 
from  any  x,  say  x1}  we  find  the  residual  to  be 

)#,  --  &2  --  #o  —  ...  --  xn  ..........  (23) 

n/   l    n   2     n   3  n 

We  now  put  e  for  the  unknown  mean  error  of  each  x.  Then, 
by  §  24,  Eq.  (7),  the  square  of  the  probable  mean  error  of  the 
linear  function  (23)  is 


This  quantity  is  an  expression  for  the  probable  value  of  the 
square  of  any  one  residual,  taken  at  random.  We  have  n  such 
squares  which,  when  equated  to  the  respective  residuals,  give  us 
n  probable  equations  of  the  form 


Ti 


60  THE   METHOD   OF  LEAST   SQUARES  [§31. 

The  sum  of  these  n  equations  gives  for  e  the  probable  value 


n-l 
In  the  example  before  us,  we  have  n==  12.     Hence 

2r2     5-38 


11  ~"   11 

and  6=±0"70. 

This  is  the  probable  value  of  the  mean  error  already  defined, 
which  the  writer  deems  the  best  to  use,  as  the  expression  of  the 
uncertainty  of  a  result.  But  it  is  quite  common  to  use  the  so- 
called  probable  error,  or  the  error  which  there  seems  to  be  an 
even  chance  of  exceeding  in  the  case  of  any  one  observation. 
Assuming  the  respective  probabilities  of  errors  of  different 
magnitudes  to  follow  the  normal  law  stated  in  §  30,  it  can  be 
shown  that  the  probable  error  is  equal  to  the  mean  error  multi- 
plied by  the  factor 

0-  =  0-6745. 

In  the  present  case,  this  gives  +  0"'47  as  the  probable  error  of 
a  single  observation. 

Finally,  the  probable  error  of  the  result  is  found  by  dividing 
the  probable  error  of  one  observation  by  the  square  root  of  the 
number  of  observations.  Thus  we  may  express  the  mean  result 
of  the  observations  given  above,  together  with  its  uncertainty, 
in  the  form 
N.RD.  of  Aldebaran  =  73°  41'  37"-04±0"-20(m.e.)  or  ±0"'14  (p.e.). 

32.  Case  of  unequal  weights. 

Let  us  now  consider  the  more  complex  case  in  which  the 
results  to  be  combined  are  of  different  weights.  As  a  numerical 
example,  we  take  the  following  six  measures  of  the  interval  of 
time  taken  by  light  in  passing  from  Fort  Myer  to  the  Washington 
Monument  and  back,  made  by  the  author  in  1882.  The  intervals 
are  expressed  in  millionths  of  a  second. 

The  weights  are  assigned  according  to  the  number  of  turns  of 
the  revolving  mirror  from  which  the  ray  was  reflected,  and  all 
other  circumstances  affecting  the  quality  of  the  result. 


§33.]  CASE   OF  UNEQUAL  WEIGHTS  61 

In  taking  a  mean  by  weights  there  is  no  need  of  multiplying 
the  whole  of  any  one  result  by  its  weight.  We  may  divide  each 
x  into  two  parts,  the  one  an  arbitrary  quantity  XQ,  the  same  for 
all,  the  other  the  difference  between  x  and  XQ,  say  A.  Then,  we 
take  the  mean  of  all  the  A's,  and  add  it  to  x0.  In  the  com- 
putation we  have  taken  x0  =  24*82,  and  multiplied  the  excess  A 
of  the  result  over  x0,  by  the  weight. 

The  products,  wA,  are  found  in  the  fourth  column,  and 
divided  by  Eio  =  30  to  form  the  weighted  mean.  In  forming 
the  residuals,  we  transfer  the  decimal  point  to  follow  the 
thousandth  of  millionths  place. 

1882.     Interval  of  time.  Weight.  wA.  r.  wr.  ivrz. 


July  24 

24-828 

4 

32 

+  0-4 

+  1-6 

i 

26 

24-828 

3 

24 

+  0-4 

+  1-2 

0 

Aug.  9 

24-822 

2 

4 

-5-6 

-11-2 

63 

10 

24-825 

5 

25 

-2-6 

-13-0 

34 

11 

24-828 

6 

48 

+  0-4 

+  2-4 

1 

29 

24-831 

6 

66 

+  3-4 

+  20-4 

69 

30 

24-827 

4 

28 

-0-6 

-  2-4 

1 

Sum,  - 

— 

30 

227 

— 

-  i-o 

169 

Mean,         -  24'8276      —  28'2 

Each  residual  is  then  multiplied  by  the  weight,  and  the  alge- 
braic sum  of  the  products,  which  should  vanish,  taken  as  a 
control.  The  sum  —1*0  is  the  remainder  neglected  in  dividing 
by  30,  the  sum  of  the  weights. 

We  next  multiply  each  wr  by  r,  so  as  to  form  wr2.  In  doing 
this,  there  is  never  any  use  in  carrying  the  product  beyond  two 
significant  figures  in  the  majority  of  the  results,  so  we  drop  the 
decimals,  and  by  adding  find 


From  this  the  probable  mean  error  is  derived  by  the  following 
investigation  : 

33.  To  find  the  probable  mean  error  when  the  weights  are  unequal. 

Let  W  be  the  sum  of  the  weights  w19w2,...wn,  and  let  rt  be 

.any  residual,  x€  —  x.     Expressing  the  latter  as  a  linear  function 


62  THE   METHOD  OF  LEAST  SQUAEES  [§  33. 

of  the  observed  quantities  by  subtracting  the  weighted  mean 
of  the  x's  from  any  one  x,  say  xt,  we  have 


w 


the  -ith  term  being  omitted  in  the  last  set  of  terms,  because 
already  included  in  the  first  term  of  the  set.  We  put  e  for 
the  mean  error  corresponding  to  weight  1.  We  shall  then  have, 
for  the  square  of  the  mean  error  of  any  one  of  the  x's,  say  XK 
by  §  26,  Eq.  (11) 

«  6 

w*' 

and  for  the  square   of   the  probable  mean  error  of  the   term 

WM 

wxK  we  shall  have 


Proceeding  as  before,  and  taking  for  K  the  successive  numbers 
1,  2,  3  ...  n,  i  alone  being  omitted,  we  find  the  square  of  the  pro- 
bable mean  error  of  the  linear  function  (25)  to  be  from  §  24, 
Eq.  (7) 


w 

The  sum  of  the  weights  w±  +  wz  ...wnis  W  — 
This  expression  therefore  reduces  to 


I 

e  ss- 


Reasoning  as  before,  this  is  the  probable  value  of  the  square 
of  the  residual  rt.     Multiplying  it  by  w{  we  have 


Putting  *'  =  !,  2,  ...  n  and  taking  the  sum  of  all  the  equations 
thus  formed,  we  have  the  probable  equation 


and  *=.  .  ...(26) 

71—1 


§34.]  PROBABLE   MEAN   ERROR  63 

The  square  root  of  this  expression  gives  the  probable  mean 
error  for  weight  1,  which,  divided  by  the  square  root  of  the  sum 
of  the  weights,  will  give  the  probable  error  of  the  result. 

In  the  example  we  have 

7i  =  7;     F=30;     2wr2 
Hence  e2  =  28'2, 


and  the  mean  result  in  units  of  '000  000  001  of  a  second  is 
Time  =  24  827-6  ±0-97  (m.e.)  or  +0'66  (p.e.). 

The  probable  error  is  therefore  less  than  the  millionth  part  of 
the  thousandth  of  a  second,  so  far  as  it  can  be  inferred  from  the 
discordance  of  the  results. 


Section  III.    Equations  of  Condition. 

34.    Elements  and  variables. 

Many  problems  of  astronomy  are  of  the  following  character : 
We  have  certain  varying  quantities  which  we  may  call 

x,  y,  z,  etc., 

of  which  we  may  determine  the  values  at  certain  moments  by 
direct  observation.  These  quantities  are  known  functions  of  the 
time  t,  and  of  other  quantities 

a,  b,  c,  etc., 

called  elements,  which  are  either  constant,  or  of  which  the 
variations  are  known  in  advance. 

x,  y,  z,  etc.,  being  functions  of  a,  b,  c,  etc.,  we  may  express 
their  relations  to  the  latter  in  the  form 

x=f(a,  b,  c,...t), (27) 

with  as  many  other  equations  as  we  have  variables  y,  z,  etc.,  to 
compute  or  observe.  We  then  have  problems  of  two  classes : 


THE   METHOD  OF  LEAST  SQUARES 


[§34. 


I.  From  known  or  assumed  values  of  the  elements  a,  b,  c,  etc., 
to  find  the  values  of  x,  y,  z,  etc.,  at  any  time. 

II.  From  a  series  of  observed  values  of  x,  y,  z,  etc.,  to  find  the 
values  of  the  elements. 

If  nearly  correct  values  of  the  elements  are  known,  we  may 
-compare  the  values  of  x,  y,  z,  etc.,  computed  from  them  with 
the  observed  values  of  those  quantities.  In  investigating  the 
relations  in  this  way  the  elements  are,  in  the  language  of  mathe- 
matics, independent  variables,  while  x,  y,  z,  etc.,  are  functions. 


FIG.  3. 

As  an  example,  let  us  take  the  case  of  an  object  P,  moving  in 
a  circle  of  radius  a  around  a  centre  0  with  a  uniform  motion. 
If  we  put  b  for  the  angle  XOW  at  a  certain  given  moment  from 
which  we  count  the  time,  which  moment  we  call  the  epoch,  and 
c  for  the  arc  through  which  the  object  moves  in  unit  of  time,  then 
the  value  of  XOP  at  any  time  t  after  the  epoch  will  be 

b  +  ct, 
and  the  rectangular  coordinates  of  P  will  be 

\ 
j  ' 


.(28) 


§  34.]  ELEMENTS  AND  VARIABLES  65 

If  a,  by  and  c  are  given,  we  may  compute  x  and  y  for  as 
many  epochs  as  we  please  by  these  equations. 

Suppose  now  that  we  can  observe  or  measure  the  coordinates 
x  and  y  at  certain  moments  tv  t2,  etc.,  after  the  epoch.  Then,  if 
a,  b,  and  c  are  known,  we  may,  by  substituting  tv  tz,  etc.,  for  t 
in  (28),  compute  x  and  y  for  the  moments  of  observation.  If  the 
computed  values  agree  with  the  observed  values,  well;  if  not, 
we  have  to  investigate  the  cause  of  the  discrepancy.  This  may 
be  either  errors  in  our  measures  of  the  coordinates,  or  errors  in 
the  values  a,  6,  and  c  used  in  the  computation.  Possibly  a  third 
cause  may  have  to  be  considered  —  error  in  the  fundamental 
hypothesis  of  uniform  circular  motion  of  P  ;  but  we  do  not 
consider  this  at  present. 

Next  take,  as  an  extreme  case,  that  in  which  the  values  of  the 
elements  a,  b,  and  c  are  entirely  unknown.  Then  we  cannot 
compute  (28)  at  all,  for  want  of  data.  What  we  have  to  do  is  to 
reverse  the  process  and  determine  a,  b,  and  c  from  the  observed 
values  of  x  and  y  at  the  known  times  tlt  tz,  etc.  If  we  call 
these  observed  values 

«i»  2/i>  a*  2/2>  etc-> 

we  shall  have  to  determine  the  values  of  a,  b,  and  c  from  the 
system  of  equations 


a  cos(b  +  ct2)  =  x.2 


Here  the  second  members  of  the  equations  are  the  observed 
values  of  x  and  y,  while  a,  b,  and  c  are  the  unknowns  to  be 
determined. 

Equations  of  this  kind  are  called  equations  of  condition, 
because  they  express  the  conditions  which  the  elements  a,  b,  and 
c  must  satisfy  in  order  that  the  results  of  computation  with 
them  .may  agree  with  observation. 

Formally,  the  unknowns  may  be  considered  as  determinable 
from  a  sufficient  number  of  independent  equations  of  the  form 

(29).     Usually  such  equations  do  not  admit  of  solution  except 

N.S.A.  E 


66  THE   METHOD  OF  LEAST  SQUARES  [§34. 

by  tentative  processes.  But  with  three  observed  values  of  x 
and  y  at  very  different  points  on  the  circle  we  may  derive 
approximate  values  of  a,  b,  and  c,  which  will  form  the  basis  for 
a  further  investigation. 

35.  Method  of  correcting  provisional  elements. 

In  most  of  the  problems  of  astronomy,  we  do  not  regard  the 
elements  themselves  as  unknown  quantities,  but  start  with 
approximate  values,  supposed  to  be  very  near  the  truth,  and 
take  as  unknowns  the  small  corrections  which  we  must  add  to 
these  assumed  or  provisional  values  in  order  to  get  the  true 
values.  The  corrections  which  these  preliminary  elements 
require  are  introduced  by  development  in  the  following  way : 

Taking  the  general  form  (27),  let 

&Q)     t?Q'     C0,  •  •  • 

be  the  provisional  values  of  the  elements  and 

Sa,  Sb,  Sc,  ... 

the  corrections  which  they  require.  Then  the  true  but  unknown 
values  of  the  elements  will  be 


.(30) 


We  substitute  these  values  in  (27)  and  develop  by  Taylor's 
theorem 


dx  .       dx    ,      dx 


(31) 
+  terms  of  the  second  and  higher  orders  in  Sa,  Sb,  etc. 

From  the  nature  of  the  case  the  provisional  values  are  quite 
arbitrary,  except  that  they  should  not  deviate  too  widely  from 
the  truth.  We  are,  therefore,  free  to  choose  their  values  so  as  to 
simplify  the  computation  whenever  this  is  practicable. 

In  practice  we  nearly  always  have  to  suppose  the  terms  of 
the  second  and  higher  orders  in  (31)  so  small  that  they  may  be 


§35.]  CORRECTING  PROVISIONAL  ELEMENTS  67 

neglected.  If  such  is  not  the  case,  it  is  commonly  easier  to 
repeat  the  computation  with  better  values  of  the  provisional 
elements  than  to  consider  the  higher  terms  in  question. 

In  the  second  member  of  (31)  the  first  term  is  the  value  of  x 
computed  with  the  assumed  values  of  the  elements.     Let  us  put 

x  comp.  ;  the  computed  value. 

x  obs.  ;  the  observed  value. 

By   taking  this  observed  value  as  the  first  member  of  (31), 
dropping  the  third  line  of  the  equation  and  transposing,  we  have 


....  ........  (32) 

Q         db0         dcQ 

In  this  equation  all  the  quantities  are  known  numerically 
except  Sa,  Sb,  and  Sc. 

Example.  The  following  coordinates  of  the  satellite  Titania 
of  the  planet  Uranus,  relative  to  the  planet,  are  derived  from 
observations  by  See  at  Washington  in  1901  : 

Time.  x.  y. 

(1)  May  13-5026  -24"'95  -  22"*05 

(2)  „      15-5007  +18-61  -26-85          (33) 

(3)  „      17-5008  +29-46  +15-03 

(4)  „      22-5014  -20  "04  -26  -67 

Let  us  as  a  first  hypothesis  assume  the  motion  in  the  apparent 
orbit  to  be  circular  and  uniform.  If  we  compute  the  polar 
coordinates,  r  (or  a)  and  0  =  b  +  ct,  from  the  above  values  of  x 
and  y  for  each  of  the  four  observations,  by  the  usual  formulae 

r  cos  0  =  x 
r  sin  $  =  2/ 

we  find  the  average  value  of  r  to  be  about  33"'08.  Also  by 
dividing  the  differences  of  the  0's  by  the  elapsed  intervals  we 
find  that  the  four  values  of  0  may  be  closely  represented  by  the 
hypothesis  that 

On  May  13-5026,       0  =  221°28' 

Daily  motion  of  0  =  c=   41    15 

We  may  take  our  initial  epoch  when  we  please  ;  generally  it 

is  best  to  take  it  near  the  mean  of  all  the  times  of  observation, 


68  THE  METHOD  OF  LEAST   SQUARES  [§35. 

so  that  the  sums  of  the  positive  and  negative  values  of  t  shall 
nearly  balance  each  other.  For  the  first  part  of  the  computation, 
however,  it  will  best  serve  our  purpose  to  take  a  moment  near 
the  first  observation,  namely  May  13'5,  as  the  epoch.  Our  values 
of  t^  t2  =  etc.  will  then  be  found  by  subtracting  this  date  from 
the  others,  and  will  be 


*!  =  0-0026; 
From  (34)  we  find 

60  =  221°  28'-£1c  =  221°  22'  ...................  (35) 

We  find  the  following  values  of  r  and  0  from  the  measures 
of  x  and  y  : 


r. 

e=b+ct. 

Diff. 

(1)  33"-30 

221°  28' 

83°  16' 

(2)  32  -66 

304  44 

82  18 

(3)  33  -07 

27     2 

206     3 

(4)  33  -36 

233     5 

We  know  that,  as  a  matter  of  fact,  the  apparent  curve  de- 
scribed by  the  satellite  is  slightly  elliptical.  But,  for  the  purpose 
of  illustration,  we  shall  find  how  nearly  the  observations  can  be 
represented  on  the  hypothesis  of  circular  and  uniform  motion. 

We  therefore  adopt  these  values  of  bQ  and  c0  : 


_  991°  99'  ^ 
—  ZZ1    LL    \ 

=   41   15  j 


(36) 

.  J 

and  we  take  a  _  33'/-Qg 

We  now  have  all  the  data  for  computing  x  and  y  from  (28) 
or  (29).  The  results,  and  the  excess  of  each  observed  coordinate 
over  that  computed,  are  found  to  be  as  follows : 

»  % 

Dates.         b  +  ct.  x  comp.  y  comp.  Ax.  Ay. 

(1)  221°  28'       -24"-79        -21-91        -0"16       -0"-14 

(2)  303   54        +18-45        -27'46        +0-16       +0-61 

(3)  2624        +29-63        +14-71        -017       +0-32 

(4)  232   40        -20-06        -26'30        +0 -02       -0-37 

Here  Ace  and  A?/  are  the  excesses  of  the  observed  values  of 
x  and  y  given  in  (33)  over  the  computed  values. 


(37) 


§35.] 


CORRECTING  PROVISIONAL   ELEMENTS 


Next  we  form  the  equations  of  condition  for  the  corrections 
from  (31).     By  differentiating  (28),  we  have 


dx 
db 


dx  _   dx 
dc~  db' 


-7   = 
da 

CM  1J 

--  = 


= 
dc       db 


.(38) 


We  now  change  our  epoch  at  pleasure.  In  forming  equations 
in  which  t  enters,  it  is  generally  convenient  to  choose  as  the 
initial  epoch  a  moment  near  the  mean  of  all  the  times  of 
observation.  In  the  present  case  we  shall  have  the  simplest 
computation  by  taking  the  moment  of  the  third  observation  as 
epoch.  Then,  dropping  useless  decimals,  the  values  of  t  are 
-4,  -2,0,  +5. 

By  using  these  four  values  of  t  in  these  equations  and  the 
values  of  a0,  60,  c0  in  (36),  we  find  four  values  of  each  coefficient, 
and  eight  equations  of  the  form  (32),  four  from  x  and  four 
from  y.  These  equations  are 


-4- 

-0-749*1 

-0-662 

+21*900 

-24-8 

+   99 

=  -0-14 

-2i 

+  0-558 

+  27-5 

-   55 

=  +0-16 

-2; 

-  0-830 

+  18-5 

-   37 

=  +0'61 

0; 

+  0-896 

-14-7 

0 

=  -0-17 

0; 

+  0-444 

+  29-6 

0 

=  +0  -32 

+  5; 

-0-607 

+  26-3 

+  132 

=  +0-02 

+  5; 

-0-795 

-20-1 

-100 

=  -0-37, 

....(39) 


These  eight  equations  have  only  three  unknowns  to  be  deter- 
mined. We  cannot  satisfy  them  all  with  any  values  of  the 
unknowns;  but  whatever  values  we  adopt,  there  will  be  out- 
standing differences  between  the  two  members  of  the  equations, 
which  we  should  make  as  small  as  possible. 

These  differences  are  what  we  have  in  §  29  called  residuals. 
They  are  functions  of  the  unknown  quantities,  and  we  seek  to 


70  THE   METHOD  OF  LEAST   SQUARES  [§35. 

determine   the   best   values   of    the    latter   from   the    principle 
developed  in  §  27  : 

The   best   values   of  the  unknown   quantities  which  can  be 
.  derived  from  a  system  of  equations  greater  in  number  than  the 
unknowns  are  those  which  make  the  sum  of  the  squares  of  the 
residuals,  multiplied  by  their  respective  weights,  a  minimum. 

36.  Conditional  and  normal  equations. 

We  have  to  show  the  simple  and  elegant  process  by  which 
values  of  the  unknowns  are  found  which  reduce  the  function  of 
the  residuals  above  defined  to  a  minimum.  For  this  purpose  let 
us  consider  the  general  case  of  a  system  of  linear  equations 
exceeding  the  unknown  quantities  in  number.  We  consider  the 
absolute  terms  or  second  members  of  the  equations  to  be  affected 
by  a  greater  or  less  probable  error,  a  judgment  which  we  express 
by  assigning  to  each  such  term  a  weight  proportional  to  the 
inverse  square  of  the  probable  error. 

Let  the  conditional  equations,  with  their  weights,  be 
^z  -f-  .  .  .  =  n^  ;     weight  = 


of  which  the  number  is  supposed  to  exceed  that  of  the 
unknowns. 

We  also  put          ±r1',     ±r2  ;     ±r3  .  .  .  ±rn 

for  the  residuals  left  when  nlt  n2,  etc.,  are  subtracted  from  the 
first  members.  Any  one  of  the  equations  may  then  be  written 
in  the  form  r<  =  aJaj  +  &<y  +  cis+...-tt,  ...................  (41) 

This  equation  gives  the  r's  as  functions  of  the  unknowns 
x,  y,  z,  etc.,  and  our  problem  is  :  What  values  of  the  unknowns 
will  make  the  function 

Q  =  wlr1*  +  w9rf+...+wn't*  ....................  (42) 

a  minimum  ?  The  required  conditions  are  that  the  derivatives 
of  Q  as  to  x,  y,  and  z,  etc.,  shall  vanish.  We  have 

mm  dp  dap 

dx     di\  dx     dr2  dx 


§36.]  CONDITIONAL  AND  NORMAL  EQUATIONS  71 


with  similar  equations  in  y,  z,  etc.     Also 

-^  =  2wjrt  =  2wi(aix  +  bty  +  .  .  . 

and,  from  (41),  fa  dr, 

_=a,;    $*-**••** 

Thus  (43)  becomes,  when  we  divide  by  2, 

^z  -f  .  .  .  )  —  w^n 


+  .....,  ..........  .  .....  .  ....................  =0. 

In  the  same  way,  using  £/  instead  of  cc  in  (43), 


+ =  0. 

Continuing  the  process,  we  shall  have  a  similar  equation  for  each 
unknown  quantity. 

The  equations  may  be  expressed  in  a  condensed  form  by  putting 

[aa]  =  w^Ojf + w2a22  +  •  •  •  +  ^nar? 

[ab]  =  w1a161  +  w2a  A  + . . .  +  wn^n 
[bb]  =  wf>*  +  wjbz2  + . . .  +  wnbn2 


[an]  =  w 
We  shall  thus  have 


.(44) 


[ac]a?  +  [be]  y  +  [cc]  z+...  =  [en] 

These  are  called  normal  equations.  The  first,  originally 
derived  by  differentiating  Q,  as  to  x,  is  called  the  normal  equation 
in  x,  because  it  is  the  one  which  determines  x,  and  so  with  the 
other  unknown  quantities. 

The  most  convenient  practical  method  of  forming  the  normal 
equations  is  to  write  under  each  conditional  equation,  or  rather 
under  each  set  of  its  coefficients,  the  product  of  the  coefficients 
into  the  weight  of  the  equation. 


72  THE   METHOD   OF  LEAST  SQUARES  [§  36. 

Another  method  is  to  multiply  all  the  terms  in  each  equation 
by  the  square  root  of  its  weight,  thus  reducing  all  the  absolute 
terms  to  weight  1. 

In  either  case,  instead  of  writing  the  unknown  quantities 
after  each  coefficient,  we  write  them  once  for  all  at  the  top  of 
the  column  of  coefficients,  as  shown  in  the  scheme  which  follows. 
This  scheme  also  shows  the  arrangement  of  the  check  against 
errors,  which  we  apply  by  putting 


SCHEME  OF  CONDITIONAL  EQUATIONS. 

x  y  z  s  n  w 

a,  b,  c,        ...       s-,  n*  m, 

(46) 
a2  b2  c2       ...       s2  n2          w, 

W2a2        w2b2         w2c2     ...     w2s2        iv2n2 
etc.          etc.          etc.      ...      etc.          etc. 

We  now  take  each  a  and  multiply  it  into  all  the  quantities  in 
the  line  below  it,  writing  the  product  in  a  horizontal  line,  thus : 

WlOJ8l 


W2a2c2      ... 
wmamcm     ...     wmamsm       wmamnm 


[aa\  [ab]  [ac]  [as]  [a?i] 

The  summation  of  the  columns  will  then  give  the  values  of 

[oa],  [ab],  [ac],  ...  [an], 
as  also  of  [as]  =  iv1a1s1  +  W2a2s2  +  ____ 

We  then  proceed  in  the  same  way  with  the  6's,  multiplying 
each  fy  ;  into  the  line  of  quantities 


Adding  the  columns  as  before,  we  shall  have  the  values  of 
[bb],  [be],  ...[6s],  [bn]. 


§  37.]  CONDITIONAL  AND  NOEMAL  EQUATIONS  73 

It  is  not  necessary  to  multiply  the  b's  into  the  wa's,  because 
the  products  have  already  been  obtained  by  multiplying  the  a's- 
into  the  w&'s  : 


and  [ab]  =  [ba]. 

If  the  computation  is  correct,  we  should  have 
[as]  =  [aa]  +  [ab]  +  [ac]  +  ...} 
[bs]  =  [ab]+[bb]+[bc]+...\  ................  (47) 

[sn]  =  [an]  +  [bn]  +  [en]  +  .  .  .  ) 

In  this  way  we  find  the  coefficients  of  all  the  normal  equations. 
Then,  by  solving  the  latter,  we  shall  have  those  values  of  the 
unknown  quantities  which  will  make  the  sum  of  the  squares  of 
the  errors  into  their  weights,  or  the  function  £2  a  minimum. 

37.  Solution  of  the  normal  equations. 

In  the  usual  computations  of  spherical  astronomy,  there  are 
seldom  more  than  three  unknown  quantities.  A  brief  indication 
of  the  practical  method  of  solution  in  this  case  will,  therefore, 
suffice.  We  take  the  coefficients  of  the  first  normal  equation  : 

[aa],  [ab],  [ac] 

multiply  them  all  by  the  successive  quotients 
[ab]      [ac]        [as] 
[aa]'    [oa]'"'[aa]' 
and   write   the   products    under    the  coefficients    of    the    other 

equations.     The  product  j=  —  ~  x  [aa]  will  be  [ab]  simply,  and  so 


need  not  be  formed,  unless  as  a  test  of  the  accuracy  of  the 
multiplier.  We  shall  thus  have  pairs  of  equations,  the  first  of 
each  pair  being  the  normal  equation  in  one  of  the  quantities 
y,  z\  the  second  the  product  of  the  equation  in  x  by  the 
appropriate  factor,  thus : 

[ab]}  [bb],  [bc],...[bs],  [bn], 
[ab][ab]     [06]  [ac]        [ab]  [as]     [ab][an] 
[aa]     '       [aa]  [aa]     '        [aa] 

Subtracting  these  from  each  other,  we  shall  have  an  equation 
from  which  x  is  eliminated. 


74  THE   METHOD  OF   LEAST  SQUARES  [§37. 

Then,  applying  the  same  process  of  eliminating  x  to  the 
remaining  normal  equations,  we  shall  have  a  set  of  equations 
between  the  unknowns  y,  z,  etc. 

Subjecting  these  equations  to  the  same  process,  we  shall  reach 
a  set  of  equations  without  x  or  y.  Going  on  in  the  same  way, 
we  at  length  reach  an  equation  with  only  one  unknown  quantity, 
say  z  of  the  form  Az  =  N 

which  gives  z  =  -j  . 

Then,  by  successive  substitution  in  the  equations  previously 
formed,  we  obtain  the  values  of  the  other  unknown  quantities. 

Example.  We  may  take  as  an  example  the  equations  (39), 
first  subjecting  them  to  a  transformation.  In  the  conditional 
equations  it  is  always  convenient  to  have  the  mean  value  of  the 
coefficients  of  any  one  unknown  not  vastly  different  from  those 
of  the  other  unknowns.  In  (39)  the  coefficients  of  Sc  have  a 
mean  value  about  100  times  as  large  as  those  of  Sa  and  30  times 
those  of  Sb.  We  may  avoid  this  inconvenience  by  using  as 
unknown  quantities 


= 


(48) 


The   substitution   of   these  expressions  will  change  the  first 
equation  into  7'5a- 


Treating  the  other  equations  in  the  same  way,  and  adding  the 
three  coefficients  of  each  equation  to  form  s  the  scheme  is  this  : 


No. 

a. 

6. 

c. 

s. 

n. 

w. 

1 

—  7'OX 

+  7% 

-8-80 

-9-0 

-0"16 

1 

2 

-6-6 

-8-3 

+  9-9 

-5-0 

-0  -14 

1 

3 

+  5-6 

+  9-2 

-5-5 

+  9'3 

+  0-16 

1-5 

4 

-8-3 

+  6-2 

-37 

-5-8 

+  0-61 

1-5 

5 

+  9-0 

-4-9 

o-o 

+  4-1 

-0-17 

1 

6 

+  4-4 

+  9-9 

o-o 

+  14-3 

+  0  -32 

1 

7 

-6-1 

+  8-8 

+  13-2 

+  15-9 

+  0-02 

1 

S 

-8-0 

-67 

-10-0 

-24-7 

-0-37 

1 

$37.]  SOLUTION   OF  NORMAL  EQUATIONS  75 

We  have  next  to  form  the  normal  equations  by  (44).  We 
multiply  all  the  terms  of  the  first  equation  by  the  first  value  of 
iva  —  a  (becauvse  w=l);  then  the  terms  of  the  second  by  the 
second  value  of  wa,  etc. 

Dropping  the  last  decimal  figure  of  the  product  as  unnecessary 
we  thus  find 

aa.  ab.  ac.  as.  an. 


56-2 

-54-8 

+  66-0 

+  67-5 

+  1-20 

43-6 

+  54-8 

-65-3 

+  33-0 

+  0-92 

47-1 

'  +  75-6 

-46-2 

+  76-5 

+  1-35 

103-4 

-74-7 

+  46-1 

+  74-7 

-7-59 

81-0 

-45-0 

o-o 

+  36-0 

-1-53 

19-4 

+  44-0 

o-o 

+  63-4 

+  1-41 

37-2 

-53-1 

-80-5 

-96-4 

-0-12 

62-4 

+  52-9 

+  79-0 

+  194-3 

+  2-92 

450-3  -0-3  -0-9         +449-0         -1-44 

The  check  against  error  is 

[aa]  +  [ab]  +  [ac]  =  [as]  , 

a  condition  which  we  find  to  be  satisfied.  Thus  the  first  normal 
equation,  or  that  in  x,  is 

450a  -  0%  -  0-90  =  -  1"'44, 

the  decimal  being  dropped  from  the  coefficient  of  x  because  it  is 
unnecessary. 

We  next  multiply  the  coefficients  by  the  respective  values 
of  wb,  omitting  the  first,  because  we  already  have  the  products 
ab.  We  thus  find 

[66]  =  543-3  ;    [be]  =  -  72-1  ;     [6s]  =  +471-0  ;     [671]  =  +  14"-34. 

We  apply  the  check 


which  comes  out  470-9  =  471-0. 

The  error  of  O'l  is  less  than  the  probable  error  from  omitted 
decimals. 

Multiplying  by  the  coefficients  we,  we  find 

[cc]  =  515-4;     [cs]=  +442-6;     [cn]=  -0"'73. 


76  THE   METHOD  OF  LEAST   SQUARES  [§37. 

The  third  check  equation 


comes  out  4424  =  442-6, 

which  is  as  near  as  could  be  expected. 

As  a  final  check,  we  multiply  each  n  by  the  corresponding- 
value  of  ws,  and  add  the  eight  products. 

The  result  is  [S7?]  =  +  12"19. 

The  check  equation 

[an]  +  [bn]  +  [en]  =  [sn] 
becomes  12"-1  7  =  12"-19, 

of  which  the  error  is  as  small  as  could  be  expected. 

The  normal  equations  to  which  we  are  thus   led,  omitting 
unnecessary  decimals,  are  : 


-     0-90+449=-   T' 
-0-3  +  543      -   721   +  47l=+14-34h  ..........  (49) 

-0-9-   72-1    +515      +442=-   0  - 

It  is  unnecessary  to  write  the  coefficients  to  the  left  of  the 
diagonal  line  [aa]...[ac];  they  are,  however,  given  for  com- 
pleteness. The  values  of  the  sums  [as]  .  .  .  [cs]  are  also  written 
in,  because  they  may  be  used  as  a  check  on  the  solution. 

To  proceed  with  the  solution  in  the  regular  way,  we  should 

multiply  fhe  first  equation  by  the  factor  ^  —  ^,  and  subtract  the 

[aa\ 

product  from  the  second  ;  then  by  the  factor  ^  —  --,  ,  and  subtract 

\aa\ 

it   from   the   third.     Thus   we   eliminate   x,  and  find  the  two 
equations  543^  _  72*10  =  +  14"-34,1 

-72-l2/+515z=-   0  73,) 

We  next  multiply  the  first  of  these  equations  by   —  -^ 
giving  _  72-i    +  100  =  -  1"'90. 


Subtracting  this  from  the  last  equation,  we  eliminate  y,  and 
have  5050  =  1-17  ...............................  (51) 

Whence  0=  +0-00232  ......................  (52) 


§38.]  SOLUTION   OF  NORMAL  EQUATIONS  77 

We  now  substitute  this  value  of  z  in  the  first  equation  (50), 
and  thus  obtain  y  ;  then  the  values  of  y  and  z  in  the  first 
equation  (49)  to  obtain  x.  The  results  are 

a=-0"-0032,      Sa=  -(T03,         | 

y=  +0-0264,      56  =+0-0088,     1  .............  (53) 

z  =  4.  o  -00232,    &  =  +  0  -000232..) 

From   the  'way   in   which   we   have   formed  the   differential 

C0efficients  dx     dx     dx 

da'    dV     dc'"" 

the  value  of  Sa  comes  out  in  seconds,  and  that  of  Sb  and  Sc  in 
arc.  We  reduce  the  values  of  the  latter  to  minutes  by  multiply- 
ing by  3438',  the  minutes  in  the  unit  radius,  and  thus  obtain 


Sc  =  +   0  -8. 

Applying  these  corrections  to  the  adopted  values  of  a,  b, 
and  c,  we  have,  for  their  definitive  values  from  all  the 

observations,  OO//AK 

a  =  66  "Uo, 

6  =  221°52'-3, 
c=   41°  15'-8. 

The  next  step  is  to  compute  the  values  of  x  and  y  from  these 
elements  for  the  dates  of  the  separate  observations,  being  careful 
to  use  the  precise  values  of  t,  rather  than  the  approximate  ones. 
Subtracting  these  xs  and  y's  from  the  observed  ones,  we  have 
the  definitive  residuals  to  be  used  in  deriving  the  probable 
errors. 

38.  Weights  of  unknown  quantities  whose  values  are  derived  from 
equations  of  condition. 

The  theory  of  errors  as  developed  in  the  preceding  section 
applies  only  to  the  probable  error  of  a  directly  observed  quantity, 
and  not  to  that  of  an  element  derived  by  the  solution  of  equations 
of  condition.  The  error  of  a  result  consequent  upon  an  error  of 
observation  may  be  smaller  or  greater  than  the  latter  to  any 
extent,  the  amount  depending  on  the  relation  of  the  result  to 


78  THE   METHOD  OF  LEAST  SQUARES  [§38, 

the  observation.  For  example,  if  AB  be  a  line,  and  C  a  distant 
object,  whose  distance  from  A  or  B  is  determined  by  measuring 
the  angles  at  A  and  B,  and  then  computing  the  sides  AC  and 
BC  of  the  triangle ;  it  is  evident  that,  the  farther  away  C  is,  the 
greater  the  effect  of  an  error  in  the  angles  upon  the  computed 
distance.  The  effect  of  an  error  in  B  may  be  imagined  by 
supposing  that,  while  the  line  AB  and  the  direction  AC  remain 
fixed,  the  side  BC  is  allowed  to  turn  on  B  as  a  pivot  through  an 
angle  equal  to  the  possible  error.  The  corresponding  motion  of 
the  point  C  will  be  the  effect  of  this  error  on  A  C. 


FIG.  4. 

The  great  advantage  of  the  method  of  least  squares  arises 
from  the  fact  that  it  affords  us  a  means  of  determining  not  only 
the  values  of  the  quantities  sought,  but  the  effect  of  the  probable 
errors  of  observation  upon  those  values.  The  theory  upon  which 
this  subject  rests  is  too  extended  for  development  in  the  present 
chapter;  we  must  therefore  confine  ourselves  to  a  statement  of 
method.  The  two  steps  in  the  case  are  : 

(1)  The  determination  of  the  weight  of  the  unknown  quantity. 

(2)  The  determination  of  the  mean  or  probable  error  eQ  corre- 
sponding to  weight  unity. 

The  probable  error  of  each  quantity  is  then  found  by  dividing 
e0  by  the  square  root  of  the  weight. 

There  are  different  ways  of  finding  the  weights,  of  which  the 
one  most  easily  remembered  is  the  following : 

Having  the  normal  equations  in  the  form  (45),  we  equate 
the  absolute  term  of  each  equation  to  a  literal  quantity,  say 
A,  B,  C,  etc.  The  equations  thus  appear  in  the  form 

[aa]x  +  [ab]  y  +  \ac~\z  +  ...  =  [an]  =  A 
[ab]x+[bb]y  +  [bc]z+...  =  [bn]  =  B  \ (54) 


§  39.]  WEIGHTS   OF   UNKNOWN  QUANTITIES  79 

We  then  carry  through  a  simultaneous  duplicate  solution,  one 
numerical,  the  other  in  terms  of  A,  B,  C,  etc.,  as  literal  quantities. 
The  final  values  of  the  unknown,  will  then  be,  not  only  their 
numerical  values,  but  their  expressions  as  linear  functions  of 
A,  B,  G,  etc.,  in  the  form 

x  =  a  number  =  kA  +  IB  +  .  .  . 


where  k,  I,  etc.,  will  be  numerical  coefficients. 

If  the  numerical  work  is  correct,  the  values  of  xt  y,  etc.,  found 
by  substituting  the  values  (54)  of  A,  B,  C  in  these  expressions, 
should  be  the  same  as  those  found  by  the  numerical  solution. 

In  these  last  expressions  the  diagonal  coefficients  k,  lfy  etc.,  are 
the  reciprocals  of  the  weights  of  the  corresponding  quantities. 

To  find  the  probable  errors  from  the  discrepancies  of  the 
observations  among  themselves,  we  compute  the  residual  r  of 
each  original  equation  of  condition  by  substituting  in  it  the 
values  of  the  unknown  quantities.  We  then  form  the  sum 


and  divide  it  by  n  —  m,  n  being  the  number  of  equations  and  m 
that  of  the  unknown  quantities.  The  quotient  is  the  square  of 
the  mean  error  for  weight  unity,  which,  being  divided  by  the 
square  root  of  the  final  weight  of  each  quantity,  gives  its  mean 
error. 

39.  Special  case  of  a  quantity  varying  uniformly  with  the  time. 
Let  us  apply  the  preceding  results  to  the  following  case.     We 
have  a  quantity  x,  of  whose  value  we  know  or  assume  only  that 
it  varies  uniformly  with  the  time.     We  express  this  property  by 
putting 

t,  the  time,  measured  from  an  initial  epoch  ; 

0,  the  value  of  x  at  this  epoch  ; 

y,  the  increase  of  x  in  unit  of  time. 

We  shall  then  have,  in  general, 

x  —  z-\-ty  ...............................  (55) 

When  we  know  the  values  of  z  and  y  we  can  determine  x 
at  any  time  t  by  means  of  this  equation. 


SO  THE  METHOD  OF  LEAST  SQUARES  [§39. 

Let  it  now  be  given  that,  at  the  several  epochs 

^i>    «|»    ^3>  •••  *n 

we  have  observed  the  value  of  x  to  be 

X-^  ,   X%  ,   Xg  ,  .  .  .  3?n  , 

and  that  the  problem  is,  from  these  observations,  to  derive  values 
of  z  and  y.  We  do  this  by  equating  the  values  of  x  in  the  form 
{55)  to  the  observed  values.  For  example,  at  the  time  ^  we 
have  for  the  value  of  x  from  (55) 


while  the  observed  value  is  xl.     Equating  these,  we  have 


where  the  second  member  is  the  observed  xr  Forming  a  similar 
-equation  from  each  of  the  other  observations  and  adding  the 
weights,  we  have 

x1    (weight  =  W 


xn          „          w 

We  thus  have  a  system  of  equations  of  condition  of  which 
x  and  y  are  the  unknown  quantities  to  be  determined. 

If  the  observations  were  absolutely  free  from  error,  the  values 
of  z  and  y  could  be  determined  from  any  two  of  the  equations. 
But,  as  all  the  observations  are  liable  to  error,  let  us  put 

*'i,  r2,...rn 

for  the  residual  differences  between  the  values  of  xlt  x.2,  ...  xn  as 
computed  with  any  arbitrary  values  of  z  and  T/,  and  the  observed 
values  of  xlt  x2,  .  .  .  xn.  Then,  instead  of  the  equations  (56)  having 
the  form  as  written,  they  will  have  the  form 


or  transposing  x  to  the  first  member,  the  system  of  equations 
will  become 

.(57) 


§39.]     QUANTITY   VARYING   UNIFORMLY  WITH  TIME  81 

We  now  introduce  the  same  requirement  as  in  taking  the 
mean,  namely  that  the  sum  of  the  squares  of  the  residuals  multi- 
plied by  the  weights,  or  the  value  of 

Q  =  iViHw2r22+...  +  wnrw2, (58) 

shall  be  the  least  possible.     This  requires  that  we  shall  have 

w^dr^  +  *v2r2dr2  + . . .  +  wnrndrn  =  0. 
We  have,  by  differentiating  (57), 

drl  =  dz  +  t^dy  \ 

dr.2=dz+t.2dy  \  ,...(59) 


Multiplying  these  by  the  corresponding  values  of  r  in  (57),  the, 
condition  reduces  to  the  form 

Adz+Bdy  =  0, (60) 

where  A=w^( 


4-  

In  order  that  (60)  may  be  satisfied  for  all  values  of  dz  and  dy, 
we  must  have  A=0    B  =  0 

These  equations  may  be   written   in   a   condensed    form   by 
putting  w  =  ivl  +  w.2  + . . .  +  wnt 

the  sum  of  all  the  weights : 


'^  -  ^     I (61) 

The  equations  A  =  0,  B  =  0,  then  become 


N.S.A. 


82  THE   METHOD  OF  LEAST  SQUARES  [§  39. 

These  are  the  normal  equations.     From  them  the  values  of 
z  and  y  are  derived: 


W[tt]-[t]2 

(63) 


_ 


Having  found  the  values  of  z  and  y,  that  of  x  may  be  found 
for  any  time  t  by  the  equation 

x  =  z  +  ty  ...............................  (64) 

40.  The  mean  epoch. 

The  epoch  from  which  we  count  t  is  arbitrary.  The  com- 
putation is  simplest  when  we  take  for  this  epoch  the  weighted 
mean  of  all  the  times  of  observation.  These,  counted  from  any 
arbitrary  epoch,  being  as  before,  tv  t2,  t3,  ...  tn,  the  weighted  mean 
of  all  the  times  will  be 


Now,  we  can  take  tm  as  the  epoch  quite  as  well  as  the  original 
epoch.     Putting  rv  T2, . . .  rn  for  the  times  counted  from  tm,  we 

*  1  1  WlJ 

T,  =  *,-«« 

Then,  putting  z  for  the  value  of  x  at  the  mean  epoch  tm,  we 
have  the  equations  of  condition 


(06) 


These  equations  are  of  the  same  form  as  (56),  T  being  put  for  t. 
Hence,  by  treating  them  as  we  did  (56),  we  shall  form  normal 
equations  like  (62),  except  that  T  takes  the  place  of  t.  But,  it  is 
a  fundamental  property  of  the  times  T  counted  from  their  mean 
epoch  that  their  weighted  mean  must  vanish.  In  fact 
[r]  =  wlTl  +  W2r2  +  .  .  .  4-  wnrn  =  0, 


§41.]  THE   MEAN   EPOCH  83 

as  the  reader  can  readily  show  for  himself  by  substituting  the 
values  of  T.     Hence  the  equations  (62)  take  the  simple  form 


and  the  solution  is 

[x] 


41.  The  probable  errors  of  the  unknown  quantities  may  be 
determined  in  the  following  way.  Expressing  z  and  y  as  linear 
functions  of  xl}  x.2,  ...xn,  §24,  Eq.  7  shows  the  mean  error  of  z 
to  be  given  by  the  equation 

W 2e2  =  W2e^  +  W22e./  +  . . .  +  w^n,  (68) 

€l}  €%,...  en  being  the  respective  mean  errors  of  xlt  x2,...xn. 
But,  putting  e0  for  the  probable  error  corresponding  to  weight  1 , 
we  have  e  2 

whence  wje*  =  w^. 

Thus,  from  (68),  eJ=TF» 


This  last  equation  expresses  the  conclusion:  The  probable 
error  of  the  variable  at  the  mean  epoch  is  the  same  as  if  all  the 
observations  had  been  made  at  that  epoch. 

We  have,  in  the  same  way,  from  (61)  and  (7), 

e2  of  [rx]  =  WjV  V  +  <Ta  V  +  • 


From  (67)  and  (5),  we  see  that  the  probable  error  of  y  is 
equal  to  that  of  [rx]  divided  by  [rr].     Hence 


84  THE   METHOD  OF  LEAST  SQUARES 


NOTES  AND  REFERENCES. 

GAUSS'S  memoir,  Theoria  Combinationis  Observationum,  with  its  supple- 
ments, is  a  classic  on  this  subject,  dealing  with  it  from  the  logical  and 
mathematical  point  of  view.  A  French  translation  by  BERTRAND  has  been 
published,  Methode  des  Moindres  Carres,  Paris,  1855. 

The  best  practical  method  of  arranging  computations  and  deriving  results 
is  set  forth  in  ENCKE'S  papers,  Ueber  die  Methode  der  Kleinsten  Quadrate, 
originally  published  as  supplements  to  the  Berliner  Jahrbuch  for  1834, 
1835,  1836.  These  papers  were  subsequently  reprinted  in  the  collection, 
J.  F.  ENCKE'S  Astronomische  Abhandlungen,  three  volumes,  8vo.  Berlin, 
1866. 

WRIGHT,  Treatise  on  the  Adjustment  of  Observations,  with  Applications  to 
Geodetic  and  other  Measures  of  Precision,  New  York,  1884,  is,  as  its  title 
implies,  prepared  principally  with  a  view  to  the  problems  of  Geodesy  ;  but 
the  other  applications  are  quite  fully  treated. 

A  similar  treatise  is  MERRIMAN'S  Text  Book  on  the  Method  of  Least  Squares. 

HELMERT,  Die  Ausgleichungsrechnung  nach  der  Methode  der  Kleinsten 
Quadrate,  Leipzig,  1872,  like  Wright's  work,  has  geodetic  applications  mainly 
in  view. 

KOLL,  Methode  der  Kleinsten  Quadrate,  Berlin,  1893,  is  the  most  extended 
treatise  with  which  the  writer  is  acquainted.  It  deals  almost  entirely  with 
practical  applications,  and  the  methods  of  forming,  manipulating,  and 
solving  the  equations. 

CZUBER,  Theorie  der  Beobachtungsfehler,  Leipzig,  1891,  presents  a  very 
elegant  and  attractive  exposition  of  the  entire  subject  from  the  mathe- 
matical standpoint. 

Papers  by  J.  W.  L.  GLAISHER  in  the  Monthly  Notices  R.A.S.,  vols.  xl. 
and  xli.,  deal  with  the  forms  of  the  determinants  which  implicitly  enter  into 
the  solution,  and  are  well  worthy  of  study. 

THIELE,  T.  N.,  Theory  of  Observations,  London,  Charles  and  Edwin  Layton, 
1903,  is  a  very  clear  presentation  of  the  subject  of  errors  of  observation. 

Besides  these,  there  are  papers  in  the  Astronomical  Journal  by  JACOBI 
and  others,  and  in  the  Astronomische  Nachrichten  by  numerous  writers, 
treating  various  aspects  and  phases  of  the  subject. 

The  author  of  this  Compendium  hopes  to  publish  a  comprehensive  work 
on  the  subject  about  the  end  of  1907. 


PART  II. 

THE  FUNDAMENTAL  PRINCIPLES  OF 
SPHERICAL  ASTRONOMY. 


CHAPTER   IV. 
SPHERICAL   COORDINATES. 

Section   I.     General   Theory. 

42.  The  positions  of  the  heavenly  bodies  are  defined  by  the 
values  of  coordinates  by  methods  developed  in  analytic  geometry 
of  three  dimensions.  In  astronomy  the  system  of  coordinates 
most  used  is  a  polar  one,  which,  to  distinguish  it  from  that  of 
polar  coordinates  in  a  plane,  is  commonly  known  as  spherical. 
Rectangular  coordinates  are  in  frequent  use  in  the  computations 
of  theoretical  astronomy,  but  enter  only  incidentally  into  those 
of  spherical  astronomy.  The  fundamental  elements  of  any 
system  are : 

1.  An  origin  or  point  of  reference.  The  points  principally 
used  in  astronomy  for  this  purpose  are : 

(a)  A  point  of  observation  on  the  earth's  surface  at  which  an 
observer  may  be  supposed  located.     Coordinates  referred  to  this 
origin  are  called  apparent 

(b)  The  centre   of   the   earth.     Coordinates   referred   to   this 
origin  are  called  geocentric. 

(c)  The  centre  of  the  sun.     Coordinates  referred  to  this  origin 
are  called  heliocentric. 

The  position  of  a  body  is  completely  expressed  when  the 
direction  and  length  of  the  line  from  the  origin  to  the  body  are 
given.  When  the  spherical  system  is  used,  the  length  of  the  line, 
called  the  radius  vector,  is  one  of  the  coordinates.  The  other 
two  are  commonly  angles  determining  the  direction  of  the  radius 
vector. 


88  SPHERICAL  COORDINATES  [§42. 

The  direction  of  the  radius  vector  is  expressed  by  its  relation 
to  a  fundamental  plane  through  the  origin,  which  has,  as  its 
cognate  or  determining  concept,  a  line  through  the  origin  per- 
pendicular to  it.  It  matters  not  whether  the  line  determines  the 
plane  or  the  plane  the  line.  In  either  case  the  line  is  taken 
for  the  axis  of  Z,  and  the  plane  is  that  of  a  system  of  co- 
ordinates X,  Y. 

In  Fig.  (5),  let  0  be  the  origin. 

P,  the  point  whose  position  is  to  be  defined. 

OZ,  the  axis  of  Z. 

OX,  the  adopted  axis  of  X. 


FIG.  5. 

From  P  drop  a  perpendicular  upon  the  fundamental  plane* 
meeting  the  latter  at  the  point  Q.  The  angle  XOQ,  measured 
around  0  in  a  counter-clockwise  direction,  is  one  of  those  which 
determines  the  direction  OP.  It  may  be  designated  as  the 
longitude  of  the  point  P,  except  when  another  designation  is 
applied. 

The  other  angle  is  either  QOP,  that  which  the  radius  vector 
makes  with  the  fundamental  plane,  or  ZOP,  the  angle  which  it 
makes  with  the  fundamental  axis.  These  angles  are  the  com- 
plements of  each  other. 

The  angle  QOP  is,  in  the  absence  of  a  special  designation, 
called  the  latitude  of  P.  It  is  positive  or  negative  according  to 
whether  P  is  situated  on  the  side  of  the  fundamental  plane 
toward  the  positive  direction  of  the  axis  of  Z,  or  on  the  opposite 


§44.]  GENERAL  THEORY  8£ 

side.       The    complementary    angle    ZOP    is    called    the    polar 
distance  of  P. 

We  readily  see  that  the  latitude  is  contained  between  the 
limits  +  90°  and  —  90°.  The  polar  distance,  connected  with  the 
latitude  by  the  relation 

Polar  distance  -f  latitude  =  90° 
is  always  positive,  and  varies  between  the  limits  0°  and  180°. 

43.  The  radius  vector,  longitude,  and  latitude  of  a  heavenly 
body    being   given,  its   position  is  uniquely  determined  by  the 
following   geometric   construction.      Pass    a   sphere   round   the 
origin  as  centre  with  a  radius  equal  to  the  given  radius  vector. 
Pass  through  the  axis  of  Z  a  plane  making  with  the  plane  XOZ 
a   dihedral  angle  equal  to  the  given  longitude.     In  this   plane 
measure  an  angle  ZOP  equal  to  the  given  polar  distance.     Then 
the   intersection  of  the   line  OP  with   the  sphere    will   be   the 
position   of   P,   which   will   thus   be  completely   and   uniquely 
determined. 

From  the  preceding  definitions  it  will  be  seen  that  the  longitude 
ranges,  between  0°  and  360°.  We  may,  if  we  choose,  use  negative 
longitudes,  implying  the  measurement  from  OX  in  the  negative 
or  clockwise  direction,  This  is  frequently  convenient  when  the 
longitude  exceeds  270°. 

44.  The  celestial  sphere. 

It  is  shown  in  spherical  trigonometry  that  we  may  assist  our 
conceptions  of  the  lines,  planes,  and  angles  which  enter  into  a 
trihedral  angle  by  imagining  a  sphere  having  its  centre  at  the 
vertex  of  the  angle,  and  marking  upon  its  surface  the  points  and 
circular  arcs  in  which  the  edges  and  planes  of  the  trihedral 
angle  intersect  it.  The  parts  of  the  trihedral  angle  are  thus 
represented  by  the  six  parts  of  a  spherical  triangle. 

A  similar  help  is  invoked  in  astronomy  by  introducing  the 
conception  of  the  celestial  sphere,  upon  which  we  may  conceive 
points  to  be  marked  and  circles  to  be  drawn.  This  sphere  has 
its  visible  representation  in  the  sky,  and  we  may  conceive  points 
and  circles  upon  it  as  marked  or  drawn  on  the  sky. 


90  SPHERICAL  COORDINATES  [§44. 

It  is  common  to  consider  the  celestial  sphere  as  of  infinite 
radius.  Then  every  point  with  which  we  are  concerned  may  be 
regarded  as  situated  at  its  centre.  All  lines  parallel  to  each 
other  intersect  it  in  coincident  points,  and  all  planes  parallel  to 
-each  other  in  coincident  great  circles. 

We  may  also  conceive  a  finite  sphere  of  any  size  to  be  drawn 
around  the  point  of  reference  or  origin  of  coordinates.  There 
will  then  be  a  separate  sphere  for  each  separate  origin.  So  far 
-as  results  are  concerned,  it  is  indifferent  which  system  is  adopted 
in  thought;  but  the  conception  of  an  infinite  sphere  is  the 
simpler. 

In  this  case,  the  direction  of  every  line  R  in  space  is  repre- 
sented by  the  point  Pr  at  which  it  intersects  the  sphere,  and  the 
direction  of  every  plane  L  by  the  great  circle  Lc  in  which  it  cuts 
the  sphere. 

If  P  is  perpendicular  to  L,  Pr  is  the  pole  of  Lc. 

The  angle  between  any  two  lines  R  and  R'  is  measured  by  the 
circular  arc  between  the  points  Pr  and  Pr,  where  they  intersect 
the  sphere. 

The  dihedral  angle  between  two  planes  is  equal  to  that  at 
which  the  corresponding  great  circles  intersect. 

Correlative  with  the  conception  of  a  system  of  planes  con- 
taining a  line  R  is  that  of  a  system  of  great  circles  passing 
through  the  point  Re  of  the  sphere.  The  great  circle  Lc  having 
Pr  as  its  pole  then  intersects  the  system  at  right  angles.  The 
circles  which  form  the  system  are  then  called  secondaries  to  Lc. 

Small  circles  parallel  to  a  great  one  are  called  parallels,  and 
may  be  designated  either  as  so  related  to  the  great  circle,  or  by 
any  point  through  which  they  pass. 

It  is  shown  in  Fig.  6  how  the  coordinates  which  determine  P 
as  seen  in  Fig.  5  are  represented  on  the  sphere.  X,  Y,  and  Z,  the 
axis  of  Y  being  added  to  the  system  of  Fig.  5,  are  the  points  in 
which  the  rectangular  axes  intersect  the  sphere.  They  form  the 
vertices  of  a  triangular  rectangular  spherical  triangle.  The 
great  circle  A  YX  is  that  in  which  the  fundamental  plane  of 
X  Y  intersects  the  sphere.  P  is  the  point  of  intersection  of  the 
radius  vector  of  a  heavenly  body  with  the  sphere.  ZPR  is  a 


§44.] 


THE   CELESTIAL   SPHERE 


91 


quadrant  from  Z  through  P  to  R.  The  angle  POQ,  which  we 
have  called  the  latitude  of  the  body,  is  now  represented  by 
the  arc  PR.  The  complementary  arc  ZP  represents  the  polar 
distance.  The  arc  XR  represents  the  angle  XOR,  which  we  have 
called  the  longitude  of  the  body.  It  may  be  equally  represented 


FIG.  6. 

by  the  angle  XZR  at  the  vertex  Z  or  by  the  dihedral  angle 
between  the  planes  XOZ  and  ROZ. 

The  point  Z  in  which  the  fundamental  axis  intersects  the 
sphere  is  called  the  pole. 

The  polar  distance  is,  in  the  abstract,  a  more  convenient  co- 
ordinate than  the  latitude,  because  it  is  always  positive.  Its  use 
thus  avoids  the  danger  of  a  mistake  from  assigning  a  wrong 
algebraic  sign,  which  is  incident  to  the  use  of  latitudes.  Polar 
distances  have  been  used  at  the  Greenwich  Observatory  since 
3835,  but  in  astronomical  literature  the  latitudinal  form  is 
generally  used. 

We  shall  frequently  use  the  notation 
X,  the  longitude, 
/3,  the  latitude. 

A  feature  in  which  the  longitudinal  coordinate  X  differs  from 
the  latitudinal  one,  /3,  is  in  the  effect  of  small  changes  in  its  value 
upon  the  position  of  the  point  indicated.  A  change  A/3  in  /3 


92  SPHERICAL  COORDINATES  [§44. 

always  produces  an  equal  change  in  the  apparent  position  of  the 
body,  as  seen  from  the  origin.  But,  owing  to  the  measures  of  X 
being  made  around  the  pole  as  an  axis,  the  apparent  displace- 
ment due  to  a  given  AX  is  less,  the  nearer  the  direction  of  the 
point  P  is  to  that  of  the  pole,  the  general  law  being 
Displacement  =  cos  /3AX. 

45.    Special  fundamental  planes  and  their  associated  concepts. 

The  planes  most  used  in  astronomy  as  fundamental  are  three 
in  number,  namely : 

(a)  The  plane  of  the  horizon. 
(/3)  The  plane  of  the  equator, 
(y)  The  plane  of  the  ecliptic. 

(a)  The  plane  of  the  horizon,  or  horizontal  plane,  is  defined  as 
that  which  is  perpendicular  to  the  direction  of  gravity  at  any 
place,  or  to  the  direction  of  the  plumb-line.  The  surface  of  a 
still  liquid,  say  water  or  quicksilver,  is  coincident  with  this  plane. 

The  great  circle  in  which  the  horizontal  plane  cuts  the 
celestial  sphere  is  called  the  celestial  horizon.  The  poles  of 
the  horizon  are  the  points  at  which  the  vertical  line  intersects 
the  sphere.  That  in  the  upward  direction  is  called  the  zenith, 
that  in  the  lower  direction  the  nadir. 

A  distinction  is  sometimes  made  between  the  horizontal  plane 
passing  through  the  position  of  the  observer,  and  the  parallel 
plane  passing  through  the  centre  of  the  earth.  The  first  is  called 
the  apparent ;  the  second,  the  rational  or  geocentric  horizon. 
The  distinction  is  unnecessary  in  the  horizon  on  the  infinite 
sphere,  because  the  two  planes  cut  the  sphere  in  coincident 
circles. 

Planes  containing  a  vertical  line  are  called  vertical  planes, 
and  the  corresponding  circles  of  the  sphere  vertical  circles.  All 
vertical  circles  pass  through  the  zenith  and  nadir,  and  are 
secondary  to  the  celestial  horizon. 

A  parallel  to  the  horizon  is  called  an  almucantur. 
(/5)  The  plane  of  the  equator  is  that  which  passes  through  the 
earth's  centre  at  right  angles  to  its  axis  of  rotation.     The  circle 
in   which   it    intersects    the    earth's    surface   is   the   terrestrial 


§  45.]  SPECIAL   FUNDAMENTAL  PLANES  93 

equator.     The  great  circle  in  which  it  intersects  the  celestial 
sphere  is  called  the  celestial  equator. 

The  poles  of  the  celestial  equator  are  the  points  in  which  the 
axis  of  rotation  intersects  the  sphere.  They  are  called  the 
celestial  poles,  and  are  distinguished  as  north  and  south. 
•%  Great  circles  passing  through  the  celestial  poles  are  secondary 
to  the  equator.  That  which  passes  through  the  zenith  of  a  place 
is  called  the  celestial  meridian  of  that  place,  or  the  meridian 
simply,  and  the  points  in  which  it  intersects  the  horizon  are  the 
north  and  south  points. 

The  vertical  circle  passing  through  the  zenith  at  right  angles 
to  the  meridian  is  called  the  prime  vertical.  The  points  in 
which  it  intersects  the  horizon  are  the  east  and  west  points. 

When  a  heavenly  body  reaches  the  meridian  of  a  place,  it  is 
said  to  culminate.  The  upper  culmination  is  that  through  the 
semi-meridian  which  contains  the  zenith,  the  lower  culmination 
is  that  through  the  semi-meridian  which  contains  the  nadir.  In 
astronomical  nomenclature  it  is  common  to  indicate  a  lower 
culmination  by  the  letters  S.P.,  an  abbreviation  of  sub  polo ;  the 
upper  one  by  U.C. 

(y)  The  plane  of  the  ecliptic  is  that  in  which  the  earth  moves 
around  the  sun,  allowance  being  made  for  slight  motions  of  the 
earth's  centre  perpendicular  to  it,  and  caused  by  the  action  of 
the  moon  and  planets. 

The  great  circle  in  which  the  plane  of  the  ecliptic  intersects 
the  celestial  sphere  is  called  the  ecliptic.  The  apparent  course 
which  the  sun,  to  the  observer  on  the  earth,  appears  to  describe 
around  the  celestial  sphere  in  the  course  of  a  year,  is  practically 
coincident  with  the  ecliptic,  and  is  commonly  used  to  define  it. 

Being  great  circles,  the  ecliptic  and  celestial  equator  intersect 
at  two  opposite  points.  These  points  are  called  the  equinoxes. 
That  point  at  which  the  sun  apparently  crosses  the  celestial 
equator,  moving  toward  the  north,  is  called  the  vernal  equinox ; 
the  opposite  point  at  which  it  crosses  toward  the  south  is  called 
the  autumnal  equinox. 

The  angle  at  which  the  equator  and  ecliptic  intersect  is  called 
the  obliquity  of  the  ecliptic.  It  is  equal  to  the  dihedral  angle 


94  SPHERICAL  COORDINATES  [§45. 

between  the  planes  of  the  ecliptic  and  equator,  and  to  the  arc  of 
the  sphere  between  the  poles  of  the  ecliptic  and  of  the  equator. 

The  two  opposite  points  on  the  ecliptic  90°  from  either  equinox 
are  termed  the  solstices,  and  are  the  points  where  the  sun  reaches 
its  greatest  angular  distance  from  the  equator,  north  or  south. 

Two  great  circles,  secondaries  to  the  equator,  and  at  right 
angles  to  each  other,  pass — the  one  through  the  celestial  poles 
and  equinox,  the  other  through  the  poles  and  the  solstices. 
These  are  called  colures.  That  which  contains  the  equinoxes  is 
called  the  equinoctial  colure ;  and  that  which  contains  the 
solstices  the  solstitial  colure. 

The  poles  of  the  ecliptic  lie  on  the  solstitial  colure,  and  it  may 
be  useful  to  remember  that  the  north  pole  of  the  ecliptic  is  in 
270°,  or  1 8  hours  of  R. A.  In  middle  northern  latitudes  it  is  at  a 
greater  or  less  distance  north  of  the  zenith  in  the  early  evenings  of 
autumn,  and  is  situated  in  the  constellation  Draco.  The  nearest 
conspicuous  star  is  to  Draconis  of  the  4th  or  5th  magnitude, 
about  3°  distant  from  it. 

Conversely,  the  north  celestial  pole  is  in  90°  of  longitude,  and 
is  near  the  star  oc  Ursae  Minoris,  hence  called  Stella  Polaris 
commonly  abbreviated  to  Polaris,  or  the  Pole  star.  At  present 
the  distance  is  about  1°  12'.  The  distance  is  diminishing  through 
precession,  in  consequence  of  which  the  pole  will  continually 
approach  the  star  during  the  next  two  centuries,  passing  it  in 
the  year  2102  at  a  distance  of  about  half  a  degree. 

46.  Special  systems  of  coordinates. 

Four  systems  of  spherical  coordinates  are  used  in  astronomy, 
each  having  one  of  the  three  fundamental  planes  just  described 
as  its  plane  of  reference. 

First  system :  altitude  and  azimuth.  This  has  the  horizon 
as  its  fundamental  plane.  The  spherical  coordinates  which 
determine  the  direction  of  a  heavenly  body  referred  to  this 
system  are  altitude  and  azimuth.  The  altitude  is  the  vertical 
coordinate,  and  is  the  angle  which  the  line  drawn  to  the  body 
makes  with  the  plane  of  the  horizon,  or,  on  the  sphere,  it  is  the 
arc  of  the  vertical  circle  through  the  body  P,  contained  between 


§46.]  SPECIAL   SYSTEMS   OF  COORDINATES  95 

P  and  the  horizon.  The  zenith  distance  is  the  complementary 
distance  from  the  zenith  to  P. 

The  azimuth  of  P  is  the  longitudinal  coordinate,  and  is  the 
arc  of  the  horizon  intercepted  between  the  vertical  circle  through 
it  and  the  north  or  south  point. 

In  accordance  with  the  general  system,  the  positive  direction 
in  which  azimuth  is  measured  should  be  from  the  north  point  of 
the  horizon  through  west,  south,  and  east.  But,  in  practice  it  is 
measured  either  from  the  north  or  the  south  point,  and  in  either 
direction,  east  or  west. 

Second  system :  right  ascension  and  declination.  Here  the 
axis  of  z  is  the  rotation  axis  of  the  earth,  and  the  fundamental 
plane  is  that  of  the  equator. 

The  latitudinal  coordinate  is  the  angle  which  the  radius  vector 
of  a  heavenly  body  makes  with  the  plane  of  the  equator,  and  is 
called  the  declination  of  the  body.  The  complementary  angle 
which  it  makes  with  the  axis  of  the  earth  is  called  the  north 
polar  distance  of  the  body.  When  the  centre  of  the  earth  is 
taken  as  the  origin,  the  adjective  geocentric  is  applied  to  the 
declination;  when  a  point  on  the  surface,  the  declination  and 
polar  distance  are  called  apparent. 

Through  the  position  of  a  body  and  the  two  poles  pass  a  semi- 
circle. The  angle  which  this  semicircle  makes  with  the  equinoctial 
colure,  measured  from  west  toward  east,  is  the  longitudinal  co- 
ordinate, and  is  called  the  right  ascension  of  the  body.  We  use 
the  abbreviations : 

R.  A.  =  Right  Ascension. 
Dec.  =  Declination. 
N.P.D.  =  North  Polar  Distance. 

Positions  on  the  surface  of  the  earth  are  referred  to  the 
equatorial  system.  The  astronomical  latitude  of  a  place  is  defined 
as  the  angle  which  the  plumb-line  at  that  place  makes  with  the 
plane  of  the  equator.  Since  this  line,  in  the  upward  direction, 
marks  the  zenith,  it  follows  that  the  declination  of  the  zenith  is 
equal  to  the  latitude  of  the  place.  A  corollary  readily  seen  is 
that  the  altitude  of  the  pole,  and  the  zenith  distance  of  the  point 


96  SPHERICAL  COORDINATES  [§46. 

at  which  the  celestial  equator  intersects  the  meridian,  are  both 
equal  to  the  latitude  of  the  place.  The  complement  of  the 
latitude  is  for  brevity  called  the  colatitude.  This  is  equal  to 
the  zenith  distance  of  the  pole  and  to  the  altitude  of  the  point  of 
intersection  of  the  equator  and  meridian. 

In  this  common  system  terrestrial  longitude  corresponds  to 
Right  Ascension,  since  both  are  measured  in  the  plane  of  the 
equator  or  around  the  pole. 

Declination  being  measured  north  from  the  celestial  equator, 
and  the  meridian  zenith  distance  of  the  equator  being  equal  to 
the  latitude,  it  follows  that  the  zenith  distance  of  any  object  on 
the  meridian  is  equal  to  the  latitude  of  the  place  minus  the 
declination. 

Putting,  as  usual  in  astronomy, 

<f>  =  Astronomical  latitude, 
z  —  Zenith  distance  south, 
S  =  Declination, 

we  have  the  relation  z  =  <p  —  S. 

In  astronomical  practice  is  used  : 

a,  the  Right  Ascension  ; 


o, 


the  Declination. 


Third  system  :  declination  and  hour  angle.  Here  the  axis  of 
Z  is  still  that  of  rotation  of  the  earth.  But  the  semicircles  from 
pole  to  pole  are  conceived  to  rotate  with  the  earth  and,  therefore, 
to  be  apparently  fixed.  They  are  called  hour  circles.  Evidently 
the  celestial  meridian,  as  already  defined,  is  the  hour  circle 
through  the  zenith. 

The  angle  which  the  hour  circle  through  a  body  makes  with 
the  meridian  is  called  the  hour  angle  of  the  body.  It  is  con- 
tinually increasing  toward  the  west  owing  to  the  rotation  of  the 
earth.  Hence,  for  convenience,  it  is  taken  positively  toward  the 
west,  though  this  is  contrary  to  the  mathematical  convention  we 
have  described. 

The  parallactic  angle  is  the  angle  between  the  hour  circle  and 
the  vertical  circle  through  a  body. 


§  47.]  SPECIAL  SYSTEMS   OF  COORDINATES  97 

Fourth  system:  longitude  and  latitude.  The  fourth  system 
of  axes  has  the  ecliptic  as  its  fundamental  plane. 

The  latitude  of  a  heavenly  body  is  the  angle  which  the  line 
drawn  to  it  from  the  origin  makes  with  the  ecliptic.  The  com- 
plementary angle  which  this  line  makes  with  that  to  the  north 
pole  of  the  ecliptic  is  called  the  ecliptic  polar  distance  of  the 
body. 

Of  the  secondaries  through  the  poles  of  the  ecliptic  one  passes 
through  the  vernal  equinox;  this  is  taken  as  the  initial  circle 
ZX,  Fig.  6.  Another  secondary,  at  right  angles  to  this,  is  the 
solstitial  colure. 

The  longitude  of  a  heavenly  body  is  the  angle  which  the  semi- 
circle through  it  and  the  pole  of  the  ecliptic  makes  with  the 
initial  circle  through  the  vernal  equinox. 

47.  Relations  of  spherical  and  rectangular  coordinates. 

To  every  system  of  spherical  coordinates  corresponds  a 
rectangular  system  having  the  same  origin  and  the  same  funda- 
mental plane.  The  relation  may  be  seen  in  §43,  Fig.  6.  The 
axis  of  X  is  that  passing  in  the  fundamental  plane  from  the 
origin  to  the  initial  point  from  which  the  longitudinal  coordinate 
of  the  spherical  system  is  measured.  The  axis  of  Y  is  in  the 
same  plane,  perpendicular  to  X,  its  positive  direction  being 
toward  the  point  of  which  the  longitudinal  coordinate  is  90°. 
The  axis  of  Z  is  that  perpendicular  to  the  fundamental  plane,  its 
positive  direction  being  on  the  positive  side  of  the  plane. 

The  rectangular  systems  most  in  use  correspond  to  the  second 
and  fourth  of  the  spherical  systems  just  described.  That  having 
the  equator  as  its  fundamental  plane  is  called  the  equatorial 
system',  that  having  the  ecliptic  the  ecliptical  system.  These 
two  systems  have  a  common  .Z-axis,  directed  toward  the  vernal 
equinox.  The  ^T-axis  of  the  equatorial  system  intersects  the 
sphere  at  the  celestial  pole ;  that  of  the  ecliptical  system  at 
the  pole  of  the  ecliptic.  The  3^- axis  of  the  equatorial  system 
intersects  the  celestial  equator  in  90°  of  R.A. ;  that  of  the 
ecliptical  system  intersects  the  ecliptic  in  90°  of  longitude. 

Both  of  these  F-points  are  on  the  solstitial  colure  Zy,  Fig.  7. 
N.S.A.  G 


98  SPHERICAL  COORDINATES  [§47. 

The  natural  origin  of  the  equatorial  system  is  the  centre  of 
the  earth,  and  that  of  the  ecliptical  system  the  centre  of  the  sun. 
But  we  may  transfer  either  origin  to  any  point  in  space — the 
centre  of  a  planet  or  of  a  star  for  example,  the  direction  of 
the  axes  remaining  unchanged.  The  points  in  which  the  axes 
intersect  the  infinite  celestial  sphere  will  then  remain  unchanged 
also. 

If  we  put,  for  the  spherical  coordinates  and  radius  vector 

A  =  the  longitudinal  coordinate, 
ft  =  the  latitudinal  „ 

r  =  the  radius  vector, 

we  see  from  the  construction  of  Fig.  6  that  the  corresponding 
rectangular  coordinates  in  any  system  will  be  given  by  the 
equations 

x  =  r  cos  ft  cos  X  \ 

y  =  r  cos  ft  sin  X  J (1) 

z  =  r  sin  ft  } 

In  the  equatorial  system,  which  is  the  usual  one  in  astronomy, 
the  expressions  are 

x  =  r  cos  8  cos  oc  "| 

y  =  r  cos  S  sin  a  I (2) 

z  =  r  sin  S  } 

48.  Differentials  of  the  rectangular  and  spherical  coordinates. 

The  differentials  of  x,  y,  0,  in  the  general  system,  are 
dx  =  cos  ft  cos  \dr  —  r  sin  ft  cos  \d/3  —  r  cos  /3  sin  \d\, 
dy  =  cos  ft  sin  \dr-r  sin  ft  sin  \d/3  +  r  cos  /3  cos  \d\, 
dz  =  sin  ftdr+r  cos  /3d  ft, 


or 
or         dx  =  -dr—zcos\dft  — 


=  "  dr  —  z  sin  X  dj3 + a%£X 
r 

•  —  -dr-[-r  cos  ^8rf/3 


•(3) 


§49.]     RECTANGULAR  AND   SPHERICAL  COORDINATES  99 

The  inverse  expressions  for  the  differentials  of  the  polar  in 
terms  of  the  rectangular  coordinates  are  found  by  multiplying 
these  three  equations  in  order  by  the  coefficients  in  order  of 
any  one  of  the  three  differentials,  and  adding.  Thus  to  express 

/Y>       fit  ty 

dp  we  multiply  by  -,  -,  and  -.     To  express  d/3  we  multiply  by 

—  0cosA,    —z  sin  A,    and    r  cos  /3.      For    dx   we   have   only   to 
multiply  by  —  y  and  x,  noting  that 

x  sin  A  —  y  cos  A  =  0. 
We  thus  have 

7       x  j       y  -.       z  7 
dr  —  -  dx  +  -  dy  +  -  dz 

p     p  '    p  ,        ,4) 

=  cos /3  cos  A  dx  +  cos  ft  sin  A  dy  +  sm/3dz 
r2d/3  =  —  z  cos  Xdx  —  z  sin  \dy-\-r  cos  /3dz 

or  rdfi  —  —  sin  /3 cos \dx  —  sin /3 sin  Xdy  +  cos /3< 

r2cos2/3dA  =  —ydx+xdy, 
or    r  cos  /3dA  =  —  sin  A  d#  +  cos  A  dy. 


To  form  the  expressions  for  the  special  case  of  the  equatorial 
system  we  replace  A  and  /3  by  a.  and  <$,  thus  obtaining 

.  7  dy       .       dx 

cos  oaoc  =  cos  a.  -*•  —  sin  a.  — 
?•  r 

(i /y  el  1]  ( 

dS=  — sin  ^  cos  a sin  ^  since  —  +  coS(S- 

r  r  r 

We  retain  the  cosines  of  /3  and  S  as  factors  of  dX  and  doc  in 
order  that  the  displacements  represented  by  the  products  may 
represent  arcs  of  a  great  circle  on  the  sphere.  As  already 
remarked,  the  amount  of  displacement  represented  by  a  given 
value  of  dA  and  den  increases  indefinitely  as  the  pole  is  approached. 

49.  Relations  of  the  equatorial  and  ecliptic  coordinates. 

The  relations  of  these  two  systems  may  be  seen  in  Fig.  7. 
Here  Xy  is  the  equator,  X  Y  the  ecliptic  intersecting  it  at  X,  the 
vernal  equinox,  which  represents  the  common  T-axis  of  the  two 
systems.  X  is  also  the  pole  of  the  solstitial  colure  which  is 


100  SPHERICAL  COORDINATES  [§49 

ZzYy.     The  obliquity  is  also  represented  by  either  of  the  arcs 
y  Y,  zZ,  or  by  the  angle  zXZ. 
Let  us  now  put 

x,  y,  z,  the  coordinates  of  a  body  referred  to  the  equatorial 

system ; 

X,  F,  Z,  the  coordinates  of  the  same  body  referred  to  the 
ecliptical  system ; 

e,  the  obliquity  of  the  ecliptic ; 

(oj,  X),  (y,  Y),  (z,  Z),  etc.,  the  angles  between  the  several 
axes. 


FIG.  7. 
From  the  figure  we  readily  see  that 

The  well  known  formulae  of  transformation  are : 
X  =  x  cos  (x,  X)  +  ycos(y,  X)+zcos(z,  X) 
Y=  x  cos  (x,  Y)  4-  y  cos  (y,  Y)  +  z  cos  (z,  Y) 
Z=xcos(x,  Z)  +  ycos(y,  Z)  +zcos(0,  Z) 


§51.]          EQUATORIAL  AND  ECLIPTIC  COOEDINATES  101 

and,  conversely, 

x  =  Xcos(x,  X)+Ycos(x,  Y)  +  Zcos(x,  Z)} 

y  =  Xcos(y,X)+Yco8(y,  Y)  +  Zcos(y,  Z)\  .........  (6) 

0  =  X  cos  0,  Z)+  Fcos  0,  F)  +  Zcos  (z,  Z) 

We  have  thus  the  formulae  of  transformation 


Y=     2/cose  +  0sin  e\  ........................  (7) 

Z=  —y  sine  -\-z  cos  el 
and,  conversely, 


(8) 


y=  Fcose  —  -Zsine 
z  —  F  sin  e  +  Z  cos  e 


Section  II.     Problems  and  Applications  of  the  Theory  of 
Spherical    Coordinates. 

50.  Right  Ascension  is  almost  universally  expressed  in  time — 
hours,  minutes,  and  seconds — instead  of  degrees,  etc.     The  reason 
of  this  practice  is  that  R.A.  is  determined  by  means  of  the  sidereal 
time,  on  a  system  set  forth  in  the  next  chapter. 

Time  and  arc  are  mutually  converted  by  multiplying  or 
dividing  by  15.  A  table  for  readily  effecting  this  multiplication 
or  division  is  found  in  Appendix  II. 

Tables  of  logarithms  of  the  trigonometric  functions  with  the 
argument  in  time  have  been  published,  but  are  not  in  general 
use.  When  not  at  hand,  it  is  always  easy  to  make  the  required 
conversion  of  the  R.A.  into  arc.  The  principal  applications  of 
spherical  astronomy  into  which  time  does  not  enter  may  be 
stated  in  the  form  of  the  solution  of  problems. 

51.  PROBLEM  I.     To    convert    longitude    and    Latitude   into 
right  ascension  and  declination  and  vice  versa. 

The  formulae  of  conversion  are  readily  derived  from  those  for 
the  transformation  of  rectangular  coordinates.  If  in  (8)  we 


102 


SPHERICAL   COORDINATES 


[§51. 


substitute  for  X,  Y,  and  Z  the  expressions  of  x,  y,  and  z  (1),  and 
for  x,  y,  and  z  the  corresponding  values  (2)  in  a  and  S,  r  divides 
out,  and  we  have 

cos  S  cos  oc  =  cos  &  cos  X  I 

cos  S  sin  oc  =  cos  e  cos  ft  sin  X  —  sin  e  sin  ft    \ (9) 

sin  S  =  sin  e  cos  ft  sin  X  4-  cos  €  sin  ft    | 

In  the  same  way,  from  (7), 

cos  ft  cos  X  =  cos  OL  cos  S  1 

cos  /3  sin  X  =  cos  e  cos  S  sin  oc  +  sin  e  sin  3     *•• 
sin  /3 


.(10) 


==  —  sin  e  cos  S  sin  a  +  cos  e  sin  (5  J 

These  equations  are  those  among  the 
parts  of  a  spherical  triangle.  This  triangle 
is  that  whose  vertices  are  the  two  poles 
and  the  body.  The  geometric  relations 
involved  in  the  problem  will  be  better  seen 
by  deriving  them  from  this  triangle. 

Let  P  and  C*  be  the  respective  poles  of 
the  equator  and  ecliptic,  corresponding  to  z 
and  Z  in  Fig.  7,  and  S  the  direction  of  the 
star.  Let  E  be  the  vernal  equinox  which, 
being  on  each  of  the  fundamental  great 
circles,  is  90°  from  either  pole. 

PCE    is    then    a    birectangular    spherical 
triangle,  in  which  CP  is  the  obliquity,  e. 
We  also  have 

Angle  ECS  =  X,  the  longitude  of  S  taken  negatively. 
Angle  EPS  =  oi,  the  R.A.  of  8 

Side  CBf  =  90°-#  the  ecliptic  N.P.D.  of  8. 
Side  PS  =  90°-<$,  the  N.P.D.  of  8. 


FIG.  8. 


*The  relative  situation  of  the  two  poles  and  equinox  in  this  figure  is  the 
obverse  of  that  in  Figure  7,  so  as  to  show  it  as  we  actually  see  it  in  looking  up  at 
the  sky.  In  the  preceding  figures  the  celestial  sphere  has  been  represented  as  if 
seen  from  the  outside,  in  order  to  show  more  clearly  the  geometric  relations 
involved. 


.(11) 


§51.]  PROBLEMS  OF   SPHERICAL  COORDINATES  103 

Hence,  in  the  triangle  PCS, 

angle  P  =  90°  +  a  =  B 
angle  C=9Q°-\  =  A 
angle  8=  =C 

sidePO=e  =c 

side  P$=90°  —  S  =  a 
side  OS  =  90°  —  /3  =  b  j 

We  add  the  usual  symbols  for  the  sides  and  opposite  angles  in 
order  to  facilitate  writing  the  fundamental  relations  between 
the  parts,  which  give  the  equations  (9)  and  (10). 

It  is  useful  to  note  that  one  set  of  relations  may  be  derived 
from  the  other  by  interchanging  X  with  a  and  ft  with  S,  and 
changing  e  into  —  e. 

The  numerical  solution  of  the  equations  (9)  will  give  sin  8 
and  cos$  separately,  the  agreement  of  which  will  serve  as  a 
partial  check  on  the  accuracy  of  the  computation.  To  adapt 
the  formulae  to  logarithmic  computation,  we  compute  the 
auxiliaries  m  and  M  thus : 


,     . 

mcos  M  =  cos /3  sin  A     /' 

Then  sin  S  =  m  sin (M+  eh 

cos  S  sin  oi  =  m  cos(i/+ e)  [• (13) 

cos  8  cos  oc  =  cos  ft  cos  X     J 

Note  that  in  these  equations 

m  =  ES  (distance  of  S  from  Equinox), 
M  =  angle  which  ES  makes  with  the  Ecliptic, 
M+  e  =  angle  which  ES  makes  with  the  Equator. 

In  the  inverse  solution  we  may  compute 

m  sin  N= sin  S  \  nA\ 

m  cos  N  =  cos  S  sin  oc     /' 

Then  sin  ft  =  m  sin  (N  —*)} 

cos  ft  sin  X  =  m  cos(N  —  e)  I, (15) 

cos  ft  cos  X  =  cos  8  cos  a     J 
which  we  may  use  to  compute  ft  and  X. 


104  SPHERICAL  COORDINATES  [§51. 

The  computation  may  be  made  yet  shorter,  thus:    from  the 
equations  (12)  we  have 

tanJtf=ta^f (16) 

sm  A 

by  which  we  compute  Mt  m  being  omitted.  To  find  a  we  take 
the  quotient  of  the  last  two  equations  (13),  substituting  for  m 
its  value  cos /3  sin  A -r- cos  M  from  the  second  of  (12).  Thus  we 

Fir!  V(* 

. 


The  quotient  of  the  first  two  of  (13)  then  gives 

Also  corresponding  to  (16)-(18)  we  have  the  equations 

tan  S 
"since 

cos(JVr-e)tanal.  .  ...(19) 

tan  A  =  — ^f — 

cosN 

tan  /3  =  sin  X  tan(  N  — , 

But  this  abbreviated  method  may  fail  to  give  an  accurate 
result  if  oc  or  X  is  very  near  0°  or  180°,  as  the  result  may  then 
come  out  as  the  quotient  of  two  small  quantities. 

52.  Use  of  the  Gaussian  equations. 

The  Gaussian  equations  for  the  spherical  triangle  may  also  be 
used  with  advantage  in  cases  where  the  angle  S  of  the  triangle 
GPS  is  required,  and,  in  any  case,  are  rendered  attractive  in  use 
by  their  elegance  in  form.  Instead  of  8,  its  complement  E  is 
used  :  E  =  90°  —  8.  They  are  as  follows : 

ECLIPTICAL  TO  EQUATORIAL  COORDINATES. 


§52.] 


sin(45°  - 
sin(45°- 


USE   OF  THE  GAUSSIAN   EQUATIONS 

EQUATORIAL  TO  ECLIPTICAL  COORDINATES. 

-  X)  =  cos(45°  +  ioc)sin(45°  -  J 


105 


(21) 


As  an  example  of  the  conversion,  showing  the  most  convenient 
arrangement  of  the  work,  let  us  convert  the  equatorial  co- 
ordinates of  oc  Lyrae  for  1900  into  longitude  and  latitude.  The 
data  are 

R.A. 
Dec. 


>c  Lyrae,    - 

n.         ID. 

«.=    18     33 

s. 

33-162 

=  278°    23' 

!7"-43 

» 

3=   38     41 

25  -71 

y  of  the  ecliptic, 

e=   23     27 

8-26 

Usual 

Method. 

sinoc 

9-995329171 

cos  (5 

9-8923920 

cosoc 

9-163992371 

^7w  - 

sin  3  =  m  sin  N 

97959584 

mcosN 

9-8877211™ 

tanjV 

9-90823737* 

y 

141°      0'    30"-65 

e 

23     27       8  -26 

N-e 

117     33     22-39 

sin(N-e) 

9-9477069 

logm 

9-9971663 

COS(^V—  e) 

9-665223271 

sin  ft 

9-9448732 

cos  ft  sin  X 

9-662389571 

cos  ft  cos  X 

9-056384372, 

tan  X 

0-6060052 

cos/5 

9-6753239 

tan/3 

0-2695493 

X 

283°   54'    51"-36 

fl 

61     44     16  79 

106  SPHERICAL  COORDINATES  [§  52. 

Gaussian  Method. 

Ja         139°    IT    38"-72 
45°  +  ia         184     11     38-72 


e  +  (5  62       8     33-97 

e-8       -15     14     17-45 

31       4     16  -98 

-7     37       8-72 

13     55     43  -02 

52     37       8  -72 

9-3814992 
cos  (45°  +  Ja)  9-998835471 

cos(45°-i(e  +  <5))  9-9870386 

cos  (45°  -i(>  -<&))  9-7832682 

sin  (45°  + la)  8-8641271  n 

sin  (45°  - 1  (e-  £))  9'9001518 

sin  (45°  - |/3)  sin  \(E-  X)  9'3803346n 

8-6473953n 
0-7329393 

8-7642849™ 
9-9858740n 
8-7784109 

259°   31'    17"-04 
183     26       8  -44 

sin  (45°  -1/3)  9'3876385 

cos  (45° -|^8)  9-9866552 

tan  (45°  -  J/3)  9'4009833 

45°  -$/3  14°  7  5r-62 

i/3  30  52  8  -38 

ft  61  44  16  -76 

X  283  54  51  -40 

E  82  57  25  -48 


§53.]  USE   OF  THE  GAUSSIAN   EQUATIONS  107 

The  difference  between  the  results  of  two  computations 
AX  =  0"'04  and  A/@  =  0"*03,  arises  from  the  imperfections  of  the 
logarithms,  due  to  the  neglect  of  decimals  after  the  seventh. 

As  to  length  of  computation,  although  there  are  more  lines 
of  numbers  to  be  written  when  the  Gaussian  equations  are 
used,  the  numbers  of  entries  of  logarithmic  tables  is  about  the 
same  in  the  two  methods. 


53.    Check  computations. 

It  is  desirable   that   the  accuracy  of  every  computation  be 
tested.     As  tests  of  the  above  transformations  we  have 

cos  M  cos  S  sin  oc  =  cos  (M+  e)  cos  8  sin  X 
and  cos  N  cos  j3  sin  X  =  cos  (  N  —  e)  cos  3  sin  oc 


The  following  more  complete  test  is  that  of  Tietjen.*  It 
consists  in  computing  the  differences,  generally  not  large,  X  —  oc 
and  S  —  ft,  independently  from  the  final  results,  and  comparing 
them  with  those  found  by  subtraction. 


TIETJEN'S  TEST  EQUATIONS. 

sin  (X  —  oc)  =  2  cos  oc  sec  ftm  sin  Je  sin 

=  2  cos  oc  sec  /3m  sin  Je  sin  (N— 


+  Je)  \ 
-ie)  J 


' 


The  first  equation  becomes  doubtful  as  a  test  for  large  values 
of  ft,  because  sec/3  is  then  large.  The  following  similar  ones, 
derived  by  applying  Napier's  analogies  to  the  parts  of  the 
triangle  EPS,  seem  to  be  a  little  shorter  in  computation,  and 
less  liable  to  the  above-mentioned  drawback. 

.......       ' 


*  Berliner  Jahrbiich,  1879.     OPPOLZER,  Bahnbestimmung,  1,  13. 


108  SPHERICAL  COORDINATES  [§  53. 

The  following  is  the  complete  computation  of  the  last  test : 

X  283°  54'  51"-40 

a  278  23  17  -43 

S  38  41  25  -71 

/3  61  44  16  -76 

100     25     42  -47 


X  +  a         562     18       8 -83 
X-a  5     31     34-0 

Je  11     43     34  -13 

i(*+0)  50     12     51  -24 

t(XH-oc)         281       9       4-42 
i(X-a)  2     45     46-98 

tanK<?  +  /3)  0-0794865 

cosJ(X4-oc)  9-2864548 

tanje  93171562 

sinJ(XH-a)  9'9917221-n- 
secJ(X-oc)  -0005052 

sinJ(X-oc)  8-6830975 

tanJ(£-0)  9-3093835  n 

KX-o.)  2°   45'    46"-97 

J(<5-/3)  -11     31     25-54 

X-oc  5     31     33  94  (97) 

8-P       -23       2     51  -08  (05) 

X         283     54     51  -37 
ft          61     44     16  -79 

It  will  be  seen  that  the  test  values  of  X  and  ft  agree  better 
with  the  results  of  the  usual  method  than  with  those  of  the 
Gaussian  equations. 

54.  Effect  of  small  changes  in  the  coordinates. 

Supplementary  to  this  problem  we  have  that  of  finding  the 
effect  of  small  changes  in  the  values  of  one  pair  of  coordinates 
upon  the  values  of  the  other,  and  of  converting  proper  motions 


§54.]        EFFECT  OF  CHANGES   IN  THE   COORDINATES  109 

or  differential  variations.  For  this  purpose  we  require  the  angle 
at  S  of  the  spherical  triangle  PC'S,  which  is  given  by  either 
of  the  equations 

sin  $  =  cos  X  sec  8  sin  e  =  cos  oc  sec  /3  sin  e,     (26) 

S  being  taken  between  the  limits  —  90°  and  +  90°. 

The  required  differential  coefficients  may  be  found  by  putting 
the  relations  between  the  parts  of  the  spherical  triangle  EPS 
into  one  of  the  forms  (cf.  §  6) 

OL,8=f(\,/3,  e) 

or  X,/3=/(a,  (S,  e), 

and  may  be  derived  from  the  differential  relations  given  in 
Appendix  I,  on  the  system  explained  in  §  (6).  Referring  to 
Fig.  8  and  the  conventional  notation  of  the  sides  of  the  triangle 
PES  in  (11),  we  see  that  these  forms  require  the  relations  among 
the  following  combinations  of  parts  of  the  triangle : 

Fora;  parts  6,  c,  A,  B. 
For  8',  parts  a,  b,  c,  A. 
For  X;  parts  a,  c,  A,  B. 
For  /3 ;  parts  a,  b,  c,  B. 

The  relations  between  the  differentials  of  the  parts  which 
enter  into  these  four  combinations  are  respectively 

—  sin  Cdb  +  cos  a  sin  Bdc  +  sin  b  cos  CdA  +  sin  adB  =  0, 

—  da  +  cos  Cdb  +  cos  Bdc  +  sin  c  sin  Bd A  =  0, 

—  sin  Cda  +  cos  b  sin  Adc  +  sin  b dA  +  sin  a  cos  CdB  =  0, 

cos  Cda  —  db  +  cos  Adc  +  sin  a  sin  CdB  =  0. 

In  these  general  relations  we  substitute  the  expressions  for 
the  parts  and  their  differentials  in  terms  of  oc,  X,  etc.,  as  formed 
from  (11).  We  thus  find, 

cos  SdcL  =  cos  S  cos  /3d\  —  sin  Sd/3  —  sin  8  cos  oude 

d8  =  sin  S  cos  f3d\  +  cos  Sd/3  +  sin  ode 
and,  conversely, 


};  (27) 


cos  /3d\  =  cos  8  cos  8doL+ sin  Sd8-\-  sin  ft  cos  \de 
d/3=  —  sin  8  cos  8doi + cos  SdS  —  sin  \de 


J.    (28) 


110 


SPHERICAL  COORDINATES 


[§54. 


In  using  these  equations  it  is  usual,  following  Gauss,  to  use  E, 
the  complement  of  S,  instead  of  the  latter.  To  use  E  we  have 
only  to  write  cos  E  for  sin  S  and  to  take  E  between  the  limits  0° 
and  180°. 

55.  A  more  luminous  view  of  the  problem  will  be  obtained  by 
geometric  construction.  Consider  Fig.  9  to  represent  an  infini- 
tesimal region  around  the  star  infinitely  magnified.  Let  S  be 
the  original  position  of  the  star.  Consider  the  effect  of  an 
infinitesimal  displacement  SS'  upon  its  coordinates. 


Let  SP  and  S'P  be  arcs  of  the  meridian  and  SE,  S'E  arcs 
from  the  pole  of  the  ecliptic  to  the  star.  Draw  S'T  and  S'R 
perpendicular  to  SE  and  SP,  and  TV  parallel  to  SP.  We  then 
have 

S'R  =  cos  S  Aoc 


.(«> 


The  transformation  of  AX  and  A/3  into  Aoc  and  A£  is  homo- 
logous with  the  transformation  of  rectangular  coordinates  from 


§56.]        EFFECT  OF  CHANGES   IN  THE   COORDINATES  111 

a  system  in  which  SE  is  the  axis  of  X  to  one  in  which  SP  is 
that  axis.     In  fact  we  have 


=  S'U-SV=S'TcosS-STsinS) 
SR  =  VT+TU=S'TsiuS+STcosS. 
Comparing  with  (a), 

cos  <5Aoc  =  cos  S  cos  /3AA  —  sin  $A/3, 
A(S  =  sin  8  cos  /3AX  +  cos  8  A/3, 
as  by  the  analytic  method. 

If  the  logarithms  of  the  results  are  required,  so  much  of  the 
conversion  as  does  not  contain  e  may  be  made  thus  : 
h  cos  H  =  cos  /3  AX 
ft  sin  #=  A/3 

=  h  cos  (8+H) 


.(29) 


A  similar  form  is  readily  constructed  for  the  reverse  problem. 

56.  PROBLEM  II.  Given  the  R.A.  and  Dec.  of  two  bodies,  to 
find  the  distance  between  them  and  the  position  angle  of  the 
one  relative  to  the  other. 

Let  S  and  S'  be  the  bodies  arid  P  the  pole.     The  P 

angle  PSS'  which  the  great  circle  joining  the  two  / 

bodies  makes  with   the  hour  circle   through   one  of 
them,  is  then  called  the  position  angle  of  S'  when 
referred  to  S.     It  is  counted  from  the  meridian  SP, 
passing  north  through  S,  toward  the  east.     The  arc 
SS'  joining  the  bodies  is  called  their  angular  distance,     M 
and  is  called  the  distance  simply.     In  the  spherical 
triangle  SS'P  the  angle  at  P  is  the  difference  of  the 
R.A.'s,  and  PS  and  PS'  are  the  complements  of  the      FlG  ^Q 
given  declinations.     We  use  the  notation  s  =  SS',  the 
distance  of  the  bodies ;  p,  their  position  angle.     The  fundamental 
theorems  of  spherical  trigonometry  then  give 

sin  s  sin  p  =  sin  P  sin  PS'  =  cos  S'  sin  (a'  —  a)     1 

sin  s  cos  p  =  cos  S  sin  &  —  sin  S  cos  $'  cos(a/  —  oc)| (30) 

a- 1,  ~-r  cos  s  —  sin  S  sin  &  +  cos  S  cos  S'  cos  (a/  —  a.)  I 


112  SPHERICAL  COORDINATES  [§56. 

We  may  transform  the  last  two  equations  in  the  usual  way 
for  logarithmic  computation  by  computing  m  and  M  from  the 
equations  TO8inJl/  =  sin^ 

m  cos  M  —  cos  S'  cos  (oC  —  a). 

Then  sin  s  sin  p  =  cos  tf  sin  (a/  —  a.) 

sin  s  cos  p  =  m  sin(M  —  8) 
cos  s '= m  cos(M  —  S) 

will  be  three  equations  for  computing  s  and  p,  with  a  partial 
check  on  the  accuracy  of  the  computation.  But  the  check  and 
the  third  equation  will  be  useless  if  s  is  a  small  arc,  say  less 
than  5°. 

In  the  usual  applications  of  this  problem  CL'  —  OL  and  s  are  so 
small  that  their  cosines  may  be  taken  as  unity.  We  may  then 
use  the  equations 

s  sin  p  =  (a!  —  cc)cos 


(32) 
s  cos  p  =  &  —  S 

It  is  generally  the  case  that  the  position  angle  p  is  not 
required  with  precision,  or  that,  instead  of  defining  this  angle  as 
that  at  P,  we  may  take  the  mean  of  the  angles  PSS'  and  PS'R, 
which  will  differ  little  from  the  angle  which  the  arc  SS'  makes 
with  the  hour  circle  through  its  middle  point.  In  these  cases 
we  may  derive  an  approximate  formula  applicable  to  yet  greater 
values  of  s,  as  follows  : 

Put  p*t  the  exterior  angle  PS'H. 

Then 
sins  cos  p'  =  —  sin  s  cos  PS'S  =  —  cos  8'  sin  S  +  sin  S'  cos  S  cos  («.'  —  oc). 

Putting  for  brevity  A  =  ^(OL'  —  oc),  the  last  member  of  these 
equations  may  be  written 

-cos  <$'  sin  8(cos2A  +sin2  J.)  +  sin  ^  cos  S(cos2A  -sin2  A), 

9 

whence     sin  s  cos  p'  =  cos2  A  sin(<S'  —  S)  —  sin2  A  sin(<S'  +  S), 
while  in  the  same  way  the  second  equation  (30)  gives 

sinscosj?  =  cos2^L  sin(<$'  —  <S)+sin2.4  sin  (^  +  (5). 
Taking  the  half  sum  of  these  equations  and  puttingP  = 


we  have  '  2(6'-S)  .........  (33) 


§57.]       EFFECT  OF  CHANGES   IN   THE   COOKDINATES 

We  have  also 

sin  s  sin  pf  =  cos  S  sin  (OL  —  a)  ; 

taking  the  half  sum  of  this  and  the  first  of  (30), 


113 


If  s  and  OL'  —  OL  are  each  less  than  half  a  degree,  we  may  put 
^s;  sin(oc'  —  oc)  =  oc'  —  oc  and  cos  %(p'  —  p)  =  cos  £(<$'  —  S)  =  l, 
without  serious  error.  If  s  and  a'  —  a  are  less  than  15',  the  error 
will  generally  not  exceed1  0"01.  Thus  we  shall  have  from  (33) 


and  (33a) 


8  sin  P  =  (a'  -a)cos  * 


57.  To  find  the  effect  of  small  changes  in  ex.  and  S  upon 
s  and  p,  the  last  equations  are  commonly  accurate  enough,  and 
no  distinction  will  be  necessary  between  P  and  p  so  far  as 
the  differential  values  are  concerned.  By  differentiating  (34), 
writing  p  for  P,  and  putting  for  brevity  &L  =  J((5/  +  $),  we 


sin  pds+s  cos  pdp  =  cos  S-^doi'—  doC)  —  £(«.'—  oc)sin  ^ 
cos  pds  —  s  sin  pdp  —  dS'  —  dS 

Eliminating  dp  by  multiplying  the  first  of  these  equations  by 
sinp  and  the  second  by  cos/3  and  taking  their  sum,  we  find 

ds          ds      . 


.(36) 


ds 
-— 

ds 

-p  =  -  cosp  -  Js  sin2p  tan  8l 


Multiplying  the  first  equation  (35)  by  cosp  and  the  second 
by  sin£>  and  subtracting,  we  find 


sdp         sdp 

-T^-,  =  — Y~  =  cos  p  cos  6-, 


N.S.A. 


^oc          da 

-  sin  p(l  +  Js  cos_p  tancSj" 

sdi, 

—  %scosp  tandlx 

H 


.(37) 


CHAPTER   V. 
THE   MEASURE   OF   TIME  AND  RELATED  PROBLEMS. 

Section  I.    Solar  and  Sidereal  Time. 

58.  The  main  purpose  of  a  measure  of  time  is  to  define  with 
precision    the    moment   of    a    phenomenon.       The    methods    of 
expressing  a  moment  of   time   fall   under  two   divisions :    one 
relating  to  what  in  ordinary  language  is  called  the  "time  of 
day,"  and  depending  on  the   earth's  rotation   on  its  axis ;   the 
other  on  the  count  of  days,  which  leads  us  to  the  use  of  years 
or  centuries.     In  any  case,  the  foundation  of  the  system  is  the 
earth's  rotation.     The  time  of  this  rotation  we  are  obliged,  in 
all  ordinary  cases,  to  treat  as  invariable,  for  the  reason  that  its 
change,  if  any,  is  so  minute  that  no  means  are  available  for 
determining  it  with  precision  and  certainty.     There  are  theo- 
retical reasons  for  believing  that  the  speed  of  rotation  is  slowly 
diminishing  from  age  to  age,  and  observations  of  the  moon  make 
it  probable  that  there  are  minute  changes  from  one  century  to 
another.     If  such  is  the  case  the  retardation  is  so  minute  that 
the  change  in  the  length  of  any  one  day  cannot  amount  to  a 
thousandth  of  a  second.     Yet,  by  the  accumulation  of  a  change 
even  smaller   than  this   through   an    entire   century,  the  total 
deviation  may  rise  to  a  few  seconds  and,  in  the  course  of  many 
centuries,  to  minutes. 

59.  Relations  of  the  sidereal  and  solar  day. 

In  ordinary  life  the  day  is  determined  by  the  apparent 
diurnal  motion  of  the  sun.  The  astronomical  day,  when  used 
for  the  measure  of  time,  rests  on  the  same  basis.  The  most 


§59.]  SIDEKEAL  AND   SOLAE  DAY  115 

natural  unit  of  time  would  be  that  of  one  rotation  of  the  earth 
on  its  axis.  But  owing  to  the  annual  motion  of  the  earth 
around  the  sun,  and  the  consequent  continual  change  of  the  sun's 
right  ascension,  the  solar  day  and  the  time  of  the  earth's  actual 
rotation  are  not  the  same,  the  latter  being  nearly  four  minutes 
less  than  the  former.  Hence,  the  introduction  into  astronomical 
practice  of  a  sidereal  day.  The  sidereal  day,  properly  so-called, 
is  the  time  of  the  earth's  rotation  on  its  axis,  and  is  equal  to  the 
interval  between  two  passages  of  an  equatorial  star  without 
proper  motion  over  the  meridian  of  a  place.  The  restriction  to 
an  equatorial  star  is  necessary  because,  owing  to  the  continual 
change  in  the  direction  of  the  earth's  axis,  known  as  precession, 
the  actual  interval  between  two  culminations  of  a  star  varies 
with  its  declination. 

The  sidereal  day  proper  is  not  used  in  astronomical  practice. 
Instead  of  the  passage  of  a  star  over  the  meridian,  we  take  the 
passage  of  the  vernal  equinox.  The  practical  sidereal  day  is  the 
interval  between  two  transits  of  the  equinox  over  the  same 
meridian.  It  is  divided  into  24  sidereal  hours,  and  these  into 
minutes  and  seconds  according  to  the  civil  custom.  For  0  h. 

o 

sidereal  time,  called  also  sidereal  noon,  is  taken  the  moment  of 
transit  of  the  vernal  equinox  over  the  meridian. 

Imagine  that,  at  the  moment  of  this  transit,  we  set  a  clock 
keeping  perfect  sidereal  time  at  0  h.  0  m.  0  s.,  and  compare  the 
apparent  motion  of  the  sidereal  sphere  with  the  clock.  As  the 
hour  angle  of  the  vernal  equinox  continually  increases  at  such  a 
rate  that  the  equinox  returns  to  the  meridian  in  24  sidereal 
hours,  it  increases  at  the  rate  of  15°  for  every  hour.  It  follows 
that  the  sidereal  clock,  when  correct,  marks  at  every  moment 
the  hour  angle  of  the  equinox.  Moreover,  since  the  right  ascen- 
sion of  a  star  is  equal  to  the  angle  between  the  hour  circles 
through  the  vernal  equinox  and  through  the  star,  it  follows 
that  the  clock,  at  every  moment,  shows  the  right  ascension  of  any 
star  which  is  on  the  meridian  at  that  moment.  In  other  words, 
it  continually  indicates  the  changing  right  ascension  of  the  meri- 
dian. At  the  end  of  24  sidereal  hours  the  vernal  equinox  once 
more  reaches  the  meridian  and  the  clock  once  more  marks  0  h. 


116          MEASURE  OF  TIME  AND  BELATED  PROBLEMS      [§  59. 

It  follows  from  this  that,  if  the  moment  of  culmination  of  any 
star  of  known  right  ascension  is  observed,  and  we  set  a  perfect 
sidereal  clock  at  that  moment  so  that  its  face  shall  indicate  the 
right  ascension,  the  indication  of  the  clock  will  remain  correct 
through  the  24  hours  and  will  show  the  R.A.  of  all  objects 
passing  the  meridian,  expressed  in  units  of  time.  This  is,  in 
principle,  the  way  in  which  right  ascensions  are  determined. 

Sidereal  time  is  used  in  astronomy  for  the  indication  of  the 
apparent  position  of  the  celestial  sphere.  As  a  general  measure 
of  time  the  mean  solar  day  is  used. 

The  natural  day  is  the  interval  between  two  culminations  of 
the  sun  over  the  meridian.  It  is  divided  into  hours,  minutes, 
and  seconds  of  sola,r  time.  The  time  determined  by  starting 
from  the  moment  of  a  culmination,  and  measuring  off  solar 
hours,  is  called  apparent  time.  It  is  equal  to  the  hour  angle  of 
the  sun  at  any  moment. 

Owing  to  the  unequal  motion  of  the  sun  in  right  ascension, 
arising  from  the  obliquity  of  the  ecliptic  and  the  eccentricity 
of  the  earth's  orbit,  the  days  and  hours  thus  determined  are 
of  unequal  length,  and  a  clock  would  have  to  be  continually 
changed  in  order  to  keep  apparent  time.  Hence,  this  measure 
of  time  is  entirely  out  of  use  for  astronomical  purposes,  and  is 
used  in  civil  life  only  in  regions  where  uniform  time  cannot 
be  obtained. 

Both  the  civil  and  astronomical  time  now  in  almost  universal 
use  are  measured  by  the  transits  of  a  mean  sun  over  the 
meridian.  This  is  a  fictitious  body  moving  uniformly  along  the 
equator,  at  such  a  rate  that  it  shall,  in  the  long  run,  be  as  much 
ahead  of  the  real  sun  as  behind  it.  The  interval  between  two 
consecutive  transits  of  this  body  is  called  the  mean  solar  day. 
The  corresponding  time  of  day  is  called  mean  solar  time.  The 
difference  between  mean  and  apparent  time  is  the  equation  of 
time,  which  is  given  in  the  Ephemeris  for  every  day  of  the  year. 

60.    Astronomical  mean  time. 

In  our  common  reckoning  of  time  the  day  begins  at  midnight, 
and  is  divided  into  two  parts  of  12  hours  each.  Time  thus 


§  61.]  ASTRONOMICAL  MEAN  TIME  117 

expressed  is  called  civil  time.  But  in  astronomical  usage  the 
day  begins  at  noon,  and  the  hours  are  counted  from  0  h.  to  24  h., 
from  each  noon  to  the  next.  Time  thus  expressed  is  called 
astronomical  mean  time ;  or  simply  mean  time. 

On  this  system  each  day  is  conceived  to  continue  till  noon  of 
the  day  following,  so  that,  for  example,  January  2,  9  h.  20  m. 
A.M.,  civil  time,  is,  in  astronomical  time,  January  1,  21  h.  20  m. 
The  following  precepts  for  changing  civil  to  astronomical  time, 
and  vice  versa,  are  obvious : 

If  the  civil  time  is  A.M.,  take  one  from  the  days,  add  12  to  the 
hours,  and  drop  A.M. 

//  P.M.  drop  P.M. 

In  either  case  the  result  is  astronomical  time. 

To  change  astronomical  to  civil  time  : 

If  astronomical  time  is  less  than  12  hours  write  P.M.  after  it. 

If  greater,  subtract  12  hours,  add  1  to  the  days,  and  write  A.M. 

61.    Time,  longitude,  and  hour  angle. 

Since  the  hour  angle  of  the  mean  sun  increases  by  360°  in  a 
mean  solar  day,  it  follows  that  it  increases  by 

15°  in  1  hour     "j 

15'  in  1  minute  j-  of  mean  solar  time. 

15"  in  1  second  J 

For  a  similar  reason,  the  hour  angle  of  the  vernal  equinox 
increases  at  the  rate  of 

15°  in  1  hour     } 

15'  in  1  minute  Vof  sidereal  time. 

15"  in  1  second  J 

Moreover,  as  the  earth  rotates,  mean  noon  passes  over  15°  of 
longitude  in  1  hour  of  mean  time,  and  sidereal  noon  in  1  hour 
of  sidereal  time.  Thus  we  may  say,  in  a  general  way,  that 
time,  expressed  in  hours,  minutes,  and  seconds,  may  be  changed 
into  arc  (°,  ',  ")  by  multiplying  by  15.  To  save  this  multi- 
plication it  is  common  to  express  right  ascension,  hour  angle,  and 
terrestrial  longitude  in  time.  This  is  equivalent  to  dividing  the 


118          MEASUKE   OF  TIME  AND   RELATED  PEOBLEMS      [§  61. 

circle  into  24  hours  instead  of  360°,  so  that  6  hours  make  a 
quadrant.  There  will  then  be  4  m.  in  every  degree  and  4  s.  in 
every  minute  of  arc. 

62.   Absolute  and  local  time. 

Since  noon,  or  any  other  hour  of  the  day,  travels  continuously 
round  the  world,  it  follows  that  the  moment  when  any  day  or 
year  begins  or  ends  varies  with  the  longitude  of  the  place. 
According  to  the  custom  now  generally  prevalent,  noon  of  any 
day,  say  January  1,  begins  when  the  sun  crosses  the  180th 
meridian  from  Greenwich,  and  ends  when  the  sun  gets  back  to 
that  meridian.  Hence  local  time  at  a  common  moment  may 
differ  by  any  amount  less  than  24  hours  for  two  places  on 
opposite  sides  of  the  180th  meridian.  We  must  therefore 
distinguish  between 

Absolute  time,  which  is  any  common  measure  of  time  to  be 
used  for  all  places,  and 

Local  time,  which  depends  on  the  longitude  of  the  place. 

The  daily  affairs  of  life  are  controlled  by  local  time,  which  is 
also  the  only  time  that  can  be  readily  and  directly  determined 
by  astronomical  observations.  If  we  have  to  compare  moments 
noted  at  different  places  we  must  reduce  each  moment  to  some 
common  standard  of  time  which  we  regard  as  absolute. 

The  usual  standard  of  absolute  time  is  the  local  time  of  some 
prime  meridian,  generally  that  of  Greenwich.  But  we  may 
equally  use  time  defined  without  reference  to  any  meridian ;  for 
example,  time  counted  from  the  moment  when  the  sun  crossed 
the  vernal  equinox.  Such  a  system  was  proposed  by  Sir 
John  Herschel,  but  has  never  come  into  use,  because  it  is  less 
convenient  than  Greenwich  time. 

Local  time  is  reduced  to  Greenwich  time  by  adding  the 
longitude  of  the  place  when  West;  subtracting  it  when  East; 
Greenwich  time  is  reduced  to  local  time  by  the  reverse 
operation. 

Astronomical  custom  is  divided  as  to  whether  East  or  West 
longitudes  shall  be  considered  positive ;  the  West  are  positive  in 
the  American  Ephemeris.  __  To  avoid  ambiguity  it  is  better  to 


§  63.]  ABSOLUTE   AND  LOCAL  TIME  119 

use  the  signs  E  or  W,  except  where  +  or  —  is  necessary.  In 
this  case  we  use  the  notation  : 

A  =  the  West  longitude  of  a  place  from  Greenwich  expressed 
in  time  ; 

if,  the  local  time  ; 

t,  the  Greenwich  time  ;  then  when  t'  is  given 


and  when  t  is  given,  t'  =  t  —  \ 

Moments  of  local  time  in  widely  separated  parts  of  the  world 
may  be  compared  by  reducing  each  to  Greenwich  time. 

63.   Recapitulation  and  illustration. 

The  following  is  a  recapitulation  and  statement  of  the  funda- 
mental definitions  and  propositions  relating  to  the  subject  of  time. 

I.  The  mean  sun  is  a  fictitious  body,  increasing  uniformly  in 
right  ascension  at  the  rate  of  24  hours,  or  360°,  in  a  solar  year, 
and  so  placed  that  the  true  sun  shall  on  the  average  be  as  much 
behind  it  as  ahead  of  it. 

II.  Mean  noon  at  any  place  is  the  moment  when  the  mean 
sun  crosses  the  meridian  of  that  place. 

III.  Mean  time  at  any  place  and  at  any  moment  is  the  West 
hour  angle  of  the  mean  sun  at  that  place  and  moment,  each 
15°  of  arc  counting  one  hour.     It  is  zero  at  noon,  and  may  be 
expressed  in  hours,  minutes,  and  seconds,  or  in  fractions  of  a  day, 
from  one  noon  to  the  next. 

IV.  Sidereal  noon,  or  sidereal  0  h.  at  any  place,  is  the  moment 
when  the  vernal  equinox  crosses  the  meridian  of  that  place. 

V.  Sidereal   time  at  any  moment  is  the  West  hour-angle  of 
the  vernal  equinox,  and  is  identical  with  the  right  ascension  of 
the  meridian  at  that  moment. 

VI.  Hence  the  sidereal  time  at  which  any  object  crosses  the 
meridian  is  its  right  ascension  at  the  moment  of  crossing. 

VII.  The  difference  between  mean  and  sidereal  time  at  any 
moment,  being  the   difference  between  the  hour  angles  of   the 
mean  sun  and  the  vernal  equinox  at  that  moment,  is  the  right 
ascension  of  the  mean  sun  in  the  sense 

Sid.  Time  —  Mean  Time  =  R.A.  Mean  Sun. 


120          MEASUKE   OF   TIME  AND  EELATED  PROBLEMS      [§  63. 

VIII.  Hence,  at   any  one  and  the  same  moment  of  absolute 
time  the  difference  between  sidereal  and  mean  time  is  the  same 
at  all  places  on  the  earth's  surface. 

IX.  Hence,  also,  the  sidereal  time  of  mean  noon  is  identical 
with  the  right  ascension  of  the  mean  sun  at  that  time. 

Illustration  of  the  propositions.  Let  0  be  the  centre  of  the 
earth,  and  the  plane  of  the  paper  that  of  the  equator,  seen  from 
the  north  side.  On  this  plane  project 

G,  the  position  of  Greenwich ; 

P,  the  position  of  any  other  place ; 

OS,  the  direction  of  the  mean  sun ; 

OE  the  direction  of  the  vernal  equinox. 


FIG.  11. 

The  inner  arrows  show  the  real  direction  of  rotation  of  the 
earth ;  the  outer  ones  the  apparent  direction  of  the  rotation  of 
the  celestial  sphere.  We  then  have  : 

Angle  GOP,  the  East  longitude  of  P ; 
„        GOS,  the  Greenwich  mean  time ; 
„       POS,  the  local  mean  time  of  P, 
„       GOE,  the  Greenwich  sidereal  time, 

POE,  the  local  sidereal  time  of  P 
„       SOE,  the  R.A.  of  the  mean  sun. 


§  65.]  EFFECT  OF  NUTATION  121 

64.  Effect  of  nutation. 

Since  the  equinoxial  point  does  not  move  uniformly  along  the 
equator,  we  introduce  a  fictitious  point  called  the  mean  equinox, 
which  moves  uniformly,  a  minute  gradual  increase  from  century 
to  century  excepted.  The  difference  in  R.A.  between  the  mean 
and  true  equinoxes  is  called  Nutation  in  Right  Ascension,  and 
is  given  for  every  tenth  day  in  the  Astronomical  Ephemeris. 
Its  greatest  amount  is  about  18"  or  1*20  s. 

The  R.A.  of  the  mean  sun  is  measured  from  the  actual 
equinox.  But  its  motion  can  be  uniform  only  when  measured 
from  the  mean  equinox.  Sidereal  time  is  measured  by  the 
transits  of  the  actual  equinox,  affected  by  nutation.  Hence  its 
units  are  not  perfectly  invariable.  But  since  the  irregularity 
does  not  amount  to  more  than  a  fraction  of  a  second  in  a  year, 
it  is  entirely  insensible  from  day  to  day.  Sidereal  time  being 
not  used  as  a  measure  of  time  through  long  periods,  this  irregu- 
larity causes  no  inconvenience. 

65.  The  year  and  the  conversion  of  mean  into  sidereal  time,  and 
vice  versa. 

The  solar  year  is  the  interval  between  two  passages  of  the 
mean  sun  through  the  mean  vernal  equinox.     Its  length  is 
Solar  Year  =  365-24220  days. 

It  is  evident  that  since  the  sun  and  the  equinox  are  again 
together  at  the  end  of  the  year,  the  equinox  has  made  one 
apparent  diurnal  revolution  more  than  the  sun.  Hence 

365-2422  solar  days  =  366*2422  sidereal  days. 

The  ratio  of  these  two  numbers  is  a  factor  by  which  intervals  of 
solar  time  may  be  changed  to  sidereal  time  or  vice  versa.  The 
most  convenient  form  for  using  the  factors  in  question  is  reached 
by  putting 

k-       l       •    /,'-        l 
"365-242'       "366-242* 

Then 

Sid.  Time  =  M.S.T.x(l  +  fc), 
M.S.T.  =  Sid.  Time  x  (1  -  k). 


122          MEASUEE   OF  TIME  AND  BELATED  PROBLEMS      [§  65. 

In  the  American  Ephemeris  (Appendix)  and  in  most  collections 
of  astronomical  tables,  and  in  Appendix  II.  of  the  present  work, 
tables  of  products  of  intervals  of  time  by  k  and  k'  are  given. 
They  are  based  on  the  equations 

24  h.  solar  time  =  (24  h.-f  3  m.  56'556  s.)  sid.  time, 
24  h.  sid.  time  =  (24  h.  — 3  m.  55*910  s.)  solar  time. 

The  reduction  may  be  made  by  taking  the  proportional  parts 
of  these  corrections  for  the  given  interval. 

If  tables  are  not  at  hand  and  the  conversion  is  not  required 
to  a  higher  degree  of  precision  than  01  s.,  a  sidereal  interval 
may  be  reduced  to  a  solar  one  by  the  following  rule : 

Divide  the  given  sidereal  interval  by  6,  taking  the  seconds  as 
reduced  to  decimals  of  a  minute,  and  write  the  hours  of  the 
quotient  in  the  minute  column,  and  the  minutes  in  the  seconds 
column.  Diminish  the  quotient  by  ^  of  its  amount;  the 
remainder  will  be  the  reduction. 

As  an  example  reduce  13  h.  4  m.  17*8  s.  sidereal  time  to  solar 
time  : 

h.        m.        s. 

6)13     4  17-8 
60)      -2  10-72 
+       218 

13     2     9'3   =  interval  of  M.S.T. 

For  the  reverse  reduction,  divide  the  first  quotient  by  70 
instead  of  60.  As  an  example 

h.       m.          s. 

6  )  13     2     9'3    m.  solar  interval. 
70)      +2  10-36 
1-86 

13     4  IT'S    sidereal  interval. 

In  each  example  we  have  added  hundredths  of  a  second  to  avoid 
an  accumulation  of  errors  in  the  tenths. 

The  preceding  conversion  is  only  that  of  intervals  between 
two  moments  of  the  same  day.  To  convert  an  actual  time  of 
day,  we  must  know  the  sidereal  time  of  mean  noon  of  the  day 
in  question.  This  requires  us  to  consider  the  general  method 
of  measuring  and  expressing  time  through  all  the  centuries. 


§  66.]  THE  GENERAL  MEASURE   OF  TIME  123 

Section  II.    The  General  Measure  of  Time. 

66.  In  Astronomy  time  is  commonly  treated  as  a  continually 
flowing  quantity,  which  it  really  is.  But  in  common  life  certain 
portions,  as  days,  months,  or  years,  are  counted  as  if  they  were 
separate  pieces  distinguished  by  ordinal  numbers.  For  example, 
that  year  which  began  with  the  assumed  moment  of  the  birth  of 
Christ  is  called  the  first  year,  or  the  year  1,  and  in  common  lan- 
guage, any  event  which  happened  during  that  year,  were  it  only 
the  day  after  Christ,  would  be  said  to  have  happened  in  the  year  1. 

But,  if  we  consider  time  as  continually  flowing,  and  express 
the  interval  from  Christ's  birth  until  any  moment  in  years  and 
decimals,  then  for  any  moment  during  the  first  year  the  interval 
would  be  only  a  fraction  of  a  year ;  for  example,  on  April  1 ,  it 
would  be  O2 5  y.;  or  0  year,  3  months.  Carrying  forward  the 
count  through  nineteen  centuries  we  see  that  April  1,  1900,  was 
really  only  1899'25  years  from  the  beginning  of  our  era.  In 
general,  when  time  is  measured  continuously  the  integral 
number  of  years  is  less  by  1  than  when  each  of  its  units  is 
taken  as  an  ordinal  number. 

To  avoid  the  inconvenience  thus  arising  astronomers  measure 
the  years  from  a  zero  epoch  one  year  earlier  than  the  birth  of 
Christ ;  that  is,  they  place  a  year  0  before  the  year  1,  and 
measure  from  its  beginning.  Thus,  a  moment  at  the  middle  of 
the  year  1900  would  be  designated  1900*5,  although  only  1899'5 
years  would  have  elapsed  since  the  Christian  era. 

This  system  leads  to  a  difference  of  one  year  between  the 
astronomical  notation  and  that  of  chronologists  in  designating 
dates  B.C.  The  two  systems  are  shown  graphically  as  follows, 
the  horizontal  line  representing  the  course  of  time  from  left  to 
right,  and  the  vertical  lines  marking  the  beginnings  of  the  years. 
Above  the  line  are  the  numbers  assigned  to  the  years  by  the 
notation  of  the  astronomers ;  below  it  those  of  the  civil  time  of 
the  chronologists. 


Astronomical 

-iy. 

oy- 

iy- 

ST- 

3y- 

Civil     - 

B.C.  2 

B.C.  1 

A.D.   1 

A.D.  2 

A.D.    3 

124          MEASURE   OF  TIME  AND   RELATED  PROBLEMS      [§  66. 

The  same  system  is  extended  to  the  days  of  the  year  and 
month.  Mean  noon  of  January  1  is  called  January  TO.  The 
zero  epoch  from  which  this  one  day  is  measured  is  noon  of 
December  31.  Hence  the  commencement  of  the  astronomical 
year  may  be  said  to  be  noon  of  December  31,  which  is  often 
called  January  0.  We  may  regard  December  31  as  belonging 
to  either  year.  Thus  the  moment  of  6  o'clock  P.M.,  on  1899 
December  31,  may  be  called 

either     1899,  December  31-25, 
or  1900,  January       0'25 ; 

while  6  o'clock  A.M.  of  1900,  January  1,  may  be  called 
either     1899,  December  3175, 
or  1900,  January       0'7o. 

67.    Units  of  time  :   the  day  and  year. 

The  fundamental  unit  for  measuring  long  intervals  of  time, 
when  the  greatest  precision  is  required,  is  the  mean  solar  day, 
as  already  defined.  Taking  any  fixed  date  as  a  fundamental 
epoch,  we  may  express  any  moment  in  history  by  the  number 
of  days  and  the  fraction  of  a  day  before  or  after  this  epoch. 
One  system  of  doing  this,  which  has  the  advantage  of  being 
continuous  through  all  history,  is  that  of  using  days  of  the 
Julian  period.  The  latter  is  taken  to  begin  4713  years'  before 
the  Christian  era,  and,  in  our  time, 

1900,  January  0  =  2415020  days  of  the  Julian  period. 

As  in  all  our  records  time  is  expressed  in  years,  there  is  an 
inconvenience  in  using  days  alone  in  computation.  Hence  the 
year  is  also  used  as  an  astronomical  unit  of  time,  and  that  of 
two  kinds,  the  Julian  and  the  solar. 

The  Julian  year  of  36 5  J  days  is  used  when  great  precision 
is  required.  The  number  of  Julian  years  and  solar  days  from 
any  date  is  easily  found,  due  allowance  being  made  for  the 
change  from  the  Julian  to  the  Gregorian  Calendar,  and  for  the 
fourths  of  a  day  which  enter  into  the  result. 


§  68.]  THE   SOLAR  OE  BESSELIAN   YEAR  125 

68.    The  solar  or  Besselian  year. 

The  solar  year  is  used  in  computations  relating  to  the  fixed 
stars.  It  is  introduced  and  based  on  the  following  data :  At  the 
fundamental  epoch  1900,  January  0,  Greenwich  mean  noon,  the 
R.A.  of  the  fictitious  mean  sun,  referred  to  the  mean  equinox, 
and  affected  by  aberration  was 

18  h.  38  m.  45-836  s., 
and  its  motion  in  a  Julian  year  is 

24  h.  0  m.  1-84542  s., 

with   a  minute  acceleration  through   several   centuries,  arising 
from  a  slight  acceleration  of  the  precession  of  the  equinoxes. 
Putting 

T,  the  R.A.  of  the  mean  sun  at  any  time ; 
T,  the  time  after  1900,  January  0,  Greenwich  M.  Noon, 
reckoned  in  Julian  centuries  of  36  525  days ;  we  have 

T  =  18  h.  38  m.  45-836  s.  +  8  640  184-542  s.  T+0'0929  s.  T2. 
In  astronomical  practice  we  take  for  the  beginning  of  a  solar 
year  the  moment  when 

T  =  280°  =  18h.  40m., 

this  falling  as  nearly  as  may  be  to  the  beginning  of  the 
Gregorian  civil  year.  It  will  be  seen  from  the  expression  for  T 
that  the  beginning  of  the  solar  year  1900  occurred  after  the 
fundamental  epoch  January  0  by  the  interval  necessary  for  the 
mean  sun  to  move  through  the  arc 

18  h.  40  m.-18  h.  38  m.  45'836  s.  =  74164  s. 
This  interval  in  decimals  of  a  day  is 

74-164x365-25 

86402      ~  =  0-313  52  day, 

so  that  the  solar  year  1900  began  at  1900,  January  0'313  52, 
Greenwich  M.T.,  which  is  1900,  January  0-0995,  Washington  M.T. 
This,  it  will  be  noted,  is  a  moment  of  absolute  time,  having  no 
reference  to  any  special  meridian.  The  solar  year  thus  defined 
is  sometimes  called  the  Besselian  fictitious  year,  after  Bessel, 
who  first  introduced  it  into  astronomy. 


126          MEASURE  OF  TIME  AND   RELATED  PROBLEMS      [§  68. 

The  beginnings  of  preceding  or  subsequent  years  may  be 
found  by  continual  addition  or  subtraction  of  36  5 '242 2  d.  Thus 
is  formed  the  table  for  the  present  century  found  in  Appendix  II. 

69.  We  now  return  to  the  relation  between  solar  and  sidereal 
time.  The  fundamental  quantity  on  which  this  relation  depends 
is  the  sidereal  time  of  mean  noon  of  any  date  at  any  longitude. 
This  is  the  same  as  the  right  ascension  of  the  mean  sun  at  the 
moment  of  mean  noon  on  that  longitude.  As  mean  noon  travels 
continuously  round  the  earth,  it  follows  that  the  sidereal  time, 
or  the  mean  right  ascension  in  question,  increases  continuously 
at  the  rate  of  3  m.  56*556  s.  for  each  mean  solar  day,  that  is 
for  each  apparent  revolution  of  the  mean  sun.  We'  have  also 
seen  that  the  value  of  the  quantity  in  question  for  the  funda- 
mental Greenwich  noon  on  1900,  January  0,  is  18  h.  38  m. 
45-836  s.  This  would  be  the  sidereal  time  of  mean  noon  for 
this  meridian  and  this  date,  when  referred  to  the  mean  equinox. 
But,  in  astronomical  practice,  as  we  have  already  remarked,  the 
equinox  taken  for  reference  is  the  true  equinox  of  the  date, 
which  may  vary  by  a  little  more  than  1  s.  from  the  mean  equinox. 
It  is,  therefore,  necessary  to  add  the  nutation  in  right  ascension, 
in  order  to  obtain  the  sidereal  time  of  noon.  As  the  latter  is 
given  in  the  ephemerides,  its  computation  is  not  necessary  except 
for  epochs  for  which  no  ephemerides  are  available. 


Section  III.    Problems  Involving  Time. 

70.  Problems  of  the  conversion  of  time. 
In  this  section  we  use  the  abbreviation 

S.T.M.N.  =  Sidereal  Time  of  Mean  Noon. 

The  ordinary  problems  of  conversion  of  time  are  the  first  three 
f ollowing : 

C5 

PROBLEM  I.  From  the  Greemvich  S.T.M.N.  to  find  that  of  the 
corresponding  date  on  any  other  meridian  of  West  longitude  \. 

Since  mean  noon  requires  the  mean  time  X  to  move  over 
longitude  X,  the  G.S.T.  required  for  the  motion  will  be  X  changed 


70.]          PROBLEMS   OF  THE   CONVERSION   OF  TIME  127 

to  sidereal  time.  But  the  local  S.T.  will  be  less  than  the 
Greenwich  S.T.  by  X.  Hence  the  local  S.T.M.N.  will  be  greater 
than  the  G.S.T.M.N.  by  the  reduction  of  X  from  mean  to  sidereal 
time,  or  S.T.M.N.  =  G.S.T.M.N.  +  k\. 

The  quantity  k\  may  be  taken  from  any  table  for  the  con- 
version of  mean  time  into  sidereal  time.  In  the  precept  X  must 
be  taken  positively  toward  the  West. 

PROBLEM  II.     To  convert  mean  time  into  sidereal  time. 

The  study  of  the  following  examples  will  render  a  rule 
unnecessary : 

Convert  1905,  Jan.  4,  8  h.  49  m.  26'36  s.  M.T.  of  Mt.  Hamilton, 
Cal.  (Long.  =  8  h.  6  m.  35  s.  W.)  into  sidereal  time. 

The  S.T.  of  the  given  moment  is  equal  to  the  S.T.M.N.  plus 
the  interval  since  M.N.  (M.T.)  reduced  to  sidereal  time.  The 
first  of  these  quantities  is  G.S.T.M.N.  +  k\  (Prob.  I.)  and  the 
second  is  M.T.  (l+/c).  We  take  from  the  ephemeris 

Greenwich  S.T.M.N.  1905,  Jan.  4  18  53  41S;85 

Reduction  to  Mt.  Hamilton  k A  (Prob.  I.)  1  19-93 

Mt.  Hamilton  S.T.M.N.  18  55  1-78 

Mt.  Hamilton  mean  time,  as  given  8  49  2 6 -36 

Reduction  to  sidereal  time    -  1  26 '97 

Mt.  Hamilton  Sidereal  Time  3  45  55-11 

Another  method  of  solution,  which  is  sometimes  more  con- 
venient, especially  when  only  an  approximate  result  is  wanted, 
makes  use  of  the  mean  time  of  sidereal  0  h.,  found  on  P.  III. 
of  each  month  of  the  Ephemeris.  The  subtractive  reduction  of 
this  M.T.  of  Sid.  0  h.  to  any  longitude  is  found  by  reducing  the 
West  longitude  from  sidereal  to  mean  time.  Thus  the  above 
example  may  be  worked  as  follows  : 

Red.  of  X=  8  h.  6  m.  35  s.  to  M.T.  0      "l  19S;72 

G.M.T.  of  Sid.  0  h.  (Ephemeris),  Jan.  4  55  27'97 

Mount  Hamilton  M.T.S.  Oh.  54  8-25 

Given  mean  time,  Jan   4  8     49  26'36 

Interval  in  mean  time  3     45  18' 11 

Red.  to  sidereal  time  -.-- ':v  -   .           •->,,)•  -U    0     00  37'01 


Sidereal  time       -         -         -  -        --    3     45     55-12 


128          MEASUEE   OF  TIME   AND   EELATED  PEOBLEMS      [§70. 

If  the  given  moment  of  mean  time  is  before  sidereal  0  h.  of 
the  same  days,  the  sidereal  0  h.  of  the  day  preceding  should 
be  used. 

PROBLEM  III.     To  convert  sidereal  time  into  mean  time. 

'Subtract  from  the  S.T.  the  S.T.M.N.,  and  we  have  the  sidereal 
interval  since  mean  noon.  Convert  this  into  mean  time,  and  the 
result  will  be  the  corresponding  mean  time. 

Reversing  the  example  of  Problem  II.  we  have : 

h.        m.  s. 

Given  sidereal  time       -  -       3     45     55*11 

S.T.M.N.      -  -     18     55       1-78 


Sidereal  interval  since  noon  -       8     50     53 '33 

Reduction  to  solar  time         -         -  -    1     26*97 


Mean  solar  time   -  8     49     26-36 

71.  Related  problems. 

PROBLEM  IV.  The  right  ascension  of  a  body  being  given,  to 
find  its  hour-angle  at  a  given  moment  of  mean  time,  and 
vice  versa. 

From  definitions  already  given  it  follows  that  the  hour-angle 
of  a  body  is  the  difference  between  its  right  ascension  and  that 
of  the  meridian.  But  the  latter  is  equal  to  the  sidereal  time. 
Hence,  putting 

h  —  the  West  hour-angle, 

we  have  h  =  t  —  OL, 

h  being  taken  positively  toward  the  west.     Hence  the  rule : 

Convert  the  given  mean  time  into  sidereal  time,  and  from  the 
latter  subtract  the  R.A.  The  remainder  is  the  hour-angle: 
West  when  positive ;  East  when  negative. 

In  the  converse  problem  the  hour-angle  is  given,  and  the  mean 
time  is  required. 

Since  t  =  h  +  cn, 

we  have  the  rule  : 

To  the  R.A.  add  the  hour-angle.  The  sum  is  the  sidereal  time, 
which  may  be  converted  into  mean  time. 


§71.]  RELATED   PROBLEMS  ,     129 

COR.  To  find  the  moment  at  which  a  heavenly  body  of 
known  R.A.  crosses  the  meridian,  we  have  only  to  take  its  R.A. 
as  sidereal  time,  and  convert  it  into  mean  time. 

PROBLEM  V.  To  find  the  mean  time  at  which  the  moon 
culminates  at  a  given  place,  on  a  given  day,  and  its  R.A.  and 
Dec.  at  the  moment  of  culmination. 

This  problem  cannot  be  solved  so  simply  as  that  preceding 
because  the  R.A.  of  the  moon  is  continually  changing,  and  is 
therefore  not  a  given  quantity.  What  is  given  in  the  Ephemeris 
is  the  moon's  R.A.  for  every  hour  of  G.M.T.  This  R.A.  never 
changes  by  more  than  an  hour  in  any  one  day.  Hence  if  we 
take  the  nearest  hour  of  R.A.  for  the  middle  of  the  day  and  add 
to  it  mentally  the  M.T.S.N.  and  the  West  longitude,  so  as  to  get 
the  sum  to  the  nearest  round  hour,  this  sum  will  be  the  G.M.T. 
of  culmination  at  the  local  meridian  within  at  least  one  or  two 
hours.  By  repeating  the  process,  using  minutes,  we  shall  have 
the  G.M.T.  within  5  minutes,  and  can  thus  find  the  nearest  hour 
of  G.M.T.  mentally. 

Of  course  the  hour  selected  need  not  be  the  absolutely  nearest 
one.  Near  the  half  hour,  either  the  hour  preceding  or  following, 
may  be  taken. 

For  this  selected  hour  of  G.M.T.  take  out  or  compute 
GCO,  the  moon's  R.A.  ; 

oc',  the  change  of  R.A.  for  1  in.  of  mean  time  ; 
TO,  the  local  sidereal  time. 

Were  the  selected  hour  exactly  that  of  culmination,  we  should 

have 

T0  =  a0. 

But  as  this  equation  will  not  be  satisfied,  we  must  find  a  number 
t  of  minutes  before  or  after  the  hour  at  which  the  equation 


is  satisfied,  T  being  the  local  sidereal  time.     Now,  in  one  minute 
of  mean  time  r  changes  by 

60  s.  (1+J5T)  ='60-1643  s. 

N.S.A.  I 


130          MEASURE   OF  TIME  AND   BELATED  PROBLEMS      [§71. 

Hence,  at  t  minutes  after  the  hour, 

T  =  T0  +  60'1643  sxt, 

oc  =  oc0  +  'oc  t  . 
Equating  these  values,  we  have 

4  _  (ao  ""  To)  (in  seconds) 

601  643s.  -a' 
and  , 


The  declination  may  then  also  be  interpolated  to  the  time  t  by 
the  formula  = 


When  many  culminations  are  to  be  computed  the  factor 


601643  s.-o." 
or  its  logarithm,  may  be  tabulated  for  every  0*01  s.  of  OL. 

EXAMPLE.  To  find  the  time  of  culmination  of  the  moon  on 
1907,  June  6,  at  San  Francisco,  A  =  8  h.  10  m.  West. 

Looking  at  pp.  94  and  97  of  the  Ephemeris,  we  find  by  using 
the  second  method  of  converting  the  Moon's  R.A.  as  S.T. 
into  M.T. 

19  h.  +  2  h.  +  8  h.  =  29  h.  or  5  h.  G.M.T. 

Thus  the  first  approximation  is 

5  h.  G.M.T.  or  21  h.  local  M.T. 

Now,  21  h.  M.T.  is  9  o'clock  A.M.  of  the  civil  day  next  following, 
and  if  we  wish  the  culmination  on  the  morning  of  June  6,  civil 
time  at  San  Francisco,  we  must  take  as  the  starting  point  of 
computation 

June  5,  21  h.  local  M.T.  =  June  6,  5  h.  G.M.T. 
Then,  our  second  approximation  will  be 

h.         m. 

M.T.S.N.,  Eph.  p.  94                          -       19  1 

Moon's  RA.,  Eph.  p.  97  -                           1  47 

Longitude  of  place                    -      i.  -         8  10 

4  58 


§  71.]  EELATED  PROBLEMS  131 

So  5  h.  G.M.T.  is  really  the  nearest  hour.  With  this  time  as 
argument,  we  take  from  the  Ephemeris 

oc0  =  l  h.  46  m.  49-69  s.       «.'  =  +1-9502  s. 

The  local  sidereal  time  must  be  accurately  computed.  It  is 
found  to  be 

T0  =  l  h.  45  m.  48-81  s. 

We  now  have  all  the  data  for  the  computation  of  t  and  oc. 
The  preceding  formulae  give 

t=  +1-0458  m.=  +1  m.  2'75  s., 
and  hence  the  increment  to  be  added  to  oc0  is 

Aa  =  '«.£=+  2-04  s. 
The  required  time  of  culmination  is  therefore 

1907,  June  6,  8  h.  51  m.  275  s.  A.M. 
The  right  ascension  of  the  moon  at  this  time  is 
a=l  h.  46  m.  51-73  s. 

In  some  cases,  another  approximation  will  be  found  necessary, 
if  the  greatest  accuracy  is  desired.  On  account  of  the  time 
falling  so  close  to  an  even  hour,  in  the  above  problem,  such 
further  approximation  is  not  necessary,  the  result  obtained  being 
accurate  to  the  nearest  hundredth  of  a  second. 

COK.  To  find  the  time  when  the  moon  has  a  given  geocentric 
hour-angle  h  at  a  given  place,  we  find  the  time  of  its  culmination 
over  a  meridian  whose  longitude  is  h  west  of  the  given  place,  or 
west  of  Greenwich. 


PROBLEM  VI.  The  R.A.  and  Dec.  of  a  star  being  given  to 
find  its  altitude,  azimuth,  and  parallactic  angle  at  a  given 
time. 

Let  MZPN  be  the  meridian. 
MN,  the  horizon. 
Z,  the  zenith. 
8,  the  position  of  the  star. 


132          MEASURE  OF  TIME  AND  RELATED   PROBLEMS      [§71. 

Then  in  the  spherical  triangle  PZS, 

PZ = co-latitude  of  place  =  90°  -  <p. 

PS  =  N.  P.D.  of  star  =  90°  -  S. 

SH  =  altitude  of  star  =  a. 

ZS  =  zenith  dist.  of  star  =  90°  —  a  =  z. 
Angle  at  P  =  hour  angle  =  h. 
Angle  at  Z=  azimuth  =  A. 
Angle  at  S  =  parallactic  angle  =  q. 


FIG.  12. 

The  given  parts  of  this  triangle  are  the  sides  PZ  and  PS  and 
the  included  angle  P.  Hence  the  Gaussian  equations  are  most 
convenient  when  all  three  of  the  remaining  parts  are  required. 
With  the  above  notation  the  Gaussian  equations  reduce  to 

sin  J^sin  \(A  —q)  =  cos  \h  sin  J(0~"^)l 
sin|2cosJ(J.—  <7)  =  sin  J/icos  |(0  +  <S)  I  n) 

cos  J0sin|(J.+g)  =  cos  J/icos  J(0  —  (5)  j 
cos  J0cos  %(A+q)  =  siu  JAsin  J(0  +  ^)  J 


The  azimuth  A  thus  found  will  be  counted  from  the  North 
point  toward  the  West,  and  will  be  180°  for  a  point  on  the 
meridian  south  of  the  zenith. 

As  a  check  upon  the  work  we  have 

sin  z  _  cos  S  _  cos  0 
sin  h  ~  sin  A  ~  sin  q 

The  elementary  formulae  of  spherical  trigonometry  may  also 
be  applied  to  the  problem,  and  will  be  simpler  than  the  Gaussian 


§71.] 


RELATED   PROBLEMS 


133 


formulae,  if  the   parallactic   angle   and   azimuth    are   not  both 
required.     They  become,  in  the  present  case 

cos  z  =  sin  0  sin  8  +  cos  0  cos  8  cos  h 
sin  z  sin  A  =  cos  8  sin  h 
sin  z  cos  A  =  cos  0  sin  ^  —  sin  0  cos  8  cos  /i 
sin  0  sin  q  =  cos  0  sin  h 
sin  0  cos  q  —  sin  </>  cos  8  —  cos  0  sin  8  cos  /*,  y 

Transforming  these  equations  in  the  usual  way,  we  have  the 
following  formulae  for  logarithmic  computation  : 

To  find  z  and  A. 

ksinK— cos  8  cos  h 
k  cos  K  =  sin  8 


sin  z  sin  A  =  cos  8  sin  /i 
sin  z  cos  J.  =  k  cos  i 
Or  eliminating  k,  we  may  use  the  formulae, 
tan  K= cot  8 cosh 
sin  ^  tan  h 


tan  J.  = 


cos 


tan  z  = 


cos  J. 


To  find  z  and  q. 


k'  sin  A"'  =  cos  0  cos  / 
//  cos  Kr  =  sin  ^ 

cos  z  =  V  sin  (jB 
sin  0  sin  q  =  cos  0  sin  h 
sin  2  cos  g  =  k'  cos  (If'  +  8) 
Or  by  the  briefer  formulae, 

tan  K'  =  cot  <p  cos  h, 
sin  K'  tan  /z, 


cosg 


(3) 


Respecting  the  briefer  formulae  it  is  to  be  remarked  that  they 
may  sometimes  fail  to  give  as  accurate  a  result  as  the  data 


134          MEASURE  OF  TIME  AND  RELATED  PROBLEMS      [§71. 

admit  of,  owing  to  tan  z  coming  out  as  the  quotient  of  two  small 
quantities.  This  will  commonly  be  the  case  when  A  or  q  differs 
little  from  90°  or  270°.  On  the  other  hand  the  extended 
formulae  are  always  accurate.  They  also  afford  a  partial  check 
upon  the  accuracy  of  the  computation  by  the  accordance  of  sin  z 
with  cos  z,  which  the  abbreviated  formulae  do  not. 

PROBLEM  VII.  The  altitude  or  Z.D.  of  a  knoivn  body,  and 
the  loMtude  of  the  place  being  given,  to  find  the  hour-angle  and 
the  local  time. 

The  first  of  equations  (2)  gives,  for  the  hour-angle, 

,   .  cos  z  —  sin  0  sin  S  ^   .  —  —.     n* 

cos  h  =  —  —  *—i  -  =  sec  0  sec  6  sin  a  —  tan  0  tan  S.  ...  (5) 

cos  0  cos  8 

The  second  form  will  be  most  convenient  when,  as  in  sextant 
work,  we  have  a  number  of  altitudes  of  the  same  body.  The 
value  of  sec  0  sec  S  and  of  tan  0  tan  <5  will  then  be  the  same  for 
all  the  altitudes.  After  finding  the  product  sec  0  sec  S  sin  a  in 
natural  numbers  we  subtract  tan  0  tan  3  from  it,  and  thus  have 
the  nat.  cosine  of  h,  and  can  at  once  find  h  from  a  table  of 
natural  sines  and  cosines. 

We  may  transform  the  first  value  of  cos/i  as  in  spherical 
trigonometry,  thus  : 

1  —  cos  h  _       2,  ,  _cos(0  —  <$)  —  cos  z 
l+cos/*,~~  ~ 

Putting  s=  1( 

this  equation  may  be  reduced  to 

tanH  h  = 


COS  8  COS  (S  —  z) 

Having  found  the  hour-angle  h,  the  sidereal  time  is  given  by 
the  equation  r  =  OL  +  h 

and  the  mean  time  is  then  found  by  conversion. 

This  problem  is  of  constant  application  in  navigation,  and 
tables  for  facilitating  its  computation  are  given  in  treatises  on 
navigation. 


I  71.]  BELATED   PKOBLEMS  135 

PROBLEM  VIII.  To  find  the  mean  time  of  sunrise  and  sunset 
•at  a  given  place. 

The  hour-angle  at  which  a  body  is  on  the  true  horizon  is 
called  its  semi-diurnal  arc.  It  is  found  by  putting  2  =  90° 

in  (5),  which  gives  7 

cos  h  =  —  tan  0  tan  8. 

If  in  this  formula  we  use  the  value  of  S  as  given  in  the 
•ephemeris  (the  geocentric  value),  the  result  will  be  the  geocentric 
hour-angle  at  which  the  body  is  on  the  geocentric  horizon. 
This  may  differ  from  the  geocentric  hour-angle  when  the  body 
is  apparently  on  the  sensible  horizon  owing  to  the  effect  of 
refraction  and  parallax.  Moreover,  in  the  case  of  the  sun  and 
moon,  it  is  the  rising  and  setting  of  the  upper  limb  and  not  of 
the  centre  which  is  usually  given  in  almanacs. 

Now,  when  the  upper  limb  of  the  sun  is  apparently  on  the 
horizon  it  is  really  34'  below  it,  being  elevated  by  refraction. 
The  centre  is  16'  below  the  limb,  or  50'  below  the  sensible 
horizon.  The  parallax  may  be  neglected  unless  the  result  is 
wanted  with  unusual  accuracy.  Hence  we  may  put 

0  =  90°  50', 

or  cos0=  —  O0145  ; 

.and  for  the  hour-angle, 

cos  h  =  —  (O0145  sec  <p  sec  3  +  tan  <f>  tan  S). 

Since  the  West  hour-angle  of  the  true  sun  is  the  apparent 
time,  this  equation  will  give  the  apparent  time  of  sunset,  to 
which  we  must  apply  the  equation  of  time  (given  in  the 
ephemeris)  to  obtain  the  mean  time. 

For  sunrise  we  subtract  h  from  12  h.  for  civil  time  or  from 
24  h.  for  astronomical  time,  and  apply  the  equation  of  time  as 
before. 

For  S  we  must  of  course  take  the  sun's  declination  not  for 
noon,  as  given  in  the  ephemeris,  but  for  the  time  of  sunrise  and 
sunset  itself.  The  change  in  declination  during  an  hour  will 
generally  be  unimportant,  so  that  we  may  need  only  a  rough 
.approximation  to  the  time  to  get  the  declination. 


136          MEASURE  OF  TIME  AND  RELATED  PROBLEMS      [§71. 

PROBLEM  IX.  To  find  the  time  of  rising  or  setting  of  the 
moon  on  a  given  day  at  a  given  place. 

When  the  moon  is  on  the  horizon  it  is  depressed  by  parallax 
by  a  quantity  which  averages  about  57'.  By  assuming  the 
parallax  to  have  this  constant  value  we  shall  in  our  latitudes 
rarely  be  led  into  an  error  of  more  than  20  s.  The  refraction 
elevates  the  moon  by  34',  and  its  mean  semi-diameter  is  15J'. 
Hence,  when  the  moon's  upper  limb  appears  to  coincide  with  the 
sensible  horizon  the  true  geocentric  Z.D.  of  its  centre  is  about 
89°  52J',  with  a  range  of  4'  on  each  side  of  this  mean.  The 
formula  for  the  geocentric  hour- angle  at  apparent  setting  of  the 
upper  limb  therefore  is 

cos  h  =  —  tan  0  tan  S  +  (V0022  sec  <£  sec  S. 

But  it  will  generally  happen  that,  owing  to  the  whole  disc 
of  the  moon  not  being  illuminated,  her  entire  visible  portion 
will  disappear  before  the  setting  of  her  upper  limb.  It  is 
therefore  best  to  take  the  setting  of  her  centre.  For  this  we 
shall  have 

geoc.  Z.D.  =  89°37', 

cos  h  =  -  tan  0  tan  S  +  0'0067  sec  0  sec  8. 

If  the  risings  and  settings  for  a  whole  year  are  to  be  computed 
for  some  one  place,  it  will  facilitate  the  work  to  make  a  table 
giving  the  value  of  h  from  this  formula  for  each  degree,  or  each 
10'  of  S  from  +29°  to  -29°. 

The  first  difficulty  we  meet  is  that  we  cannot  find  the  value 
of  S  until  we  have  an  approximate  time  of  the  phenomenon. 
The  computation  of  this  time,  and  of  the  final  result,  will  be 
facilitated  by  the  "Moon  Culminations"  of  the  Ephemeris. 
Here  are  given  the  local  mean  time  of  culmination  over  the 
Meridian  of  Washington,  the  R.A.  and  Dec.  of  the  moon  at  the 
moment  of  culmination,  and  the  variation  of  these  quantities  for 
one  hour  of  longitude.  By  this  is  meant  the  change  in  their 
values  while  the  moon  is  moving  from  the  meridian  of  Washing- 
ton to  the  meridian  1  h.  West  of  Washington,  supposing  the 


§  71.]  RELATED   PROBLEMS  137 

motion  of  the  moon  to  be  uniform.     For  example,  if  on  a  certain 

date  we  have 

Time  of  Transit  over  the  Meridian  of  Washington  9  h.  22-6  m., 
Change  in  1  hour  of  Longitude  2'27  m., 

the  local  mean  time  of  transit  over  the  meridian  1  hour  West  of 
Washington  will  be,  approximately,  9  h.  24*9  m.  And  the  W.M.T. 
of  this  transit  will  be  10  h.  24'9  m. 

Moreover,  it  must  be  remembered  that  whenever  the  geocentric 
west  hour- angle  of  the  moon  at  a  place  L  is  h,  then  the  moon  is 
on  the  meridian  of  a  place  in  longitude  h  west  of  L.  Hence, 
having  found  h,  let  X  be  the  longitude  of  L.  Then  when  the 
moon  is  setting  or  rising  at  L  she  is  on  the  meridian  of  a  place 
in  longitude  \  +  h  or  X  —  h  respectively.  Let  Tl  be  the  local  M.TV 
of  transit  over  this  place.  Then 

T^+h 

will  be  the  local  M.T.  at  L ;  that  is,  the  time  of  moon-set  at  L. 
For  moon-rise  h  must  be  taken  negatively. 

EXAMPLE.     To  find  the  time  of  moon-rise  and  moon-set  at 
San  Francisco,  1892,  June  1,  the  position  being 
0=4-37°  48', 
\=  +  3  h.  1-4  m.  West  of  Washington. 

From  the  Ephemeris  for  1892,  p.  388,  we  find,  for  this  date 

h.  m. 

Mean  time  of  transit  over  Washington      5     5  8 '01 
Eed.  to  San  Francisco,  3-02  h.  x  1-816  m.          5-48 

Local  M.T.  of  transit  over  San  Francisco    6       3 '49 

The  rising  and  setting  we  seek  are  those  preceding  and 
following  this  transit.  And  the  first  data  we  require  are  the 
declinations  of  the  moon  at  the  time  in  question. 

Let  us  put  r  for  the  amount  by  which  the  semi-diurnal  arc 
differs  from  90°  or  6  h.,  i.e.  h  =  9(T  +  T,  so  that  cos/i=-sinr. 
In  a  first  rude  approximation  we  may,  in  the  equation 

cos  h  =  —  sin  T  =  —  tan  0  tan  3, 
put  sin  T  =  T  and  tan  8  =  8.     This  gives 

(5  =  +0-776  8. 


138          MEASUEE   OF  TIME  AND   RELATED   PROBLEMS      [§  71. 

On  the  date  in  question  we  have  S=  +13°;  T=  4-10°=  +40  m.; 
h  =  +  6  h.  40  m.  roughly.  Thus,  as  a  first  rude  approximation, 
the  moon,  at  the  required  moments,  was  on  the  meridian 

6  h.  40  m.  E.  of  S.Fr.  ;     /.   3  h.  39  m.  E.  of  Wash. 
6  h.  40  m.  W.  of  S.Fr.;     .'.    9  h.  41  m.  W.  of  Wash. 

From  the  sixth  and  seventh  columns  of  the  Ephemeris  we 
now  find  the  declinations  more  accurately  as  follows  : 

Declination  at  transit  over  Washington    +13°  36' 

Change  for  -3*65  h.  of  longitude  -1-47 

+  9-68h.  -    2      6 


Declination  at  rising    - 
Declination  at  setting  - 


-  +  14 

-  +  1  1 


23 
30 


With  these  values  of  8  and  the  known  value  of  $,  we  now 
compute  the  accurate  values  of  the  hour  angle.  The  compu- 
tation is  as  follows  : 


Moon-rise. 

Moon-set. 

8 

+  14°  23' 

+  11°  30' 

tan<£ 

9-8897 

9-8897 

tan  8 

9-4090 

9-3085 

log(l) 

9-298771 

9-1982/1 

log  0-0067 

7-8261 

7-8261 

sec<£ 

0-1023 

0-1023 

sec  8 

0-0138 

0-0088 

log  (2) 

7-9422 

7-9372 

subt.  log 

0-0195 

0-0245 

cos^ 

9-279271 

9-1737/1 

h 

-100°  58' 

+  98°  35' 

= 

-  6  h.  43-9  m. 

+  6h.  34-3  m. 

At  the  moments  of  moon-rise  and  moon-set,  therefore,  the 
moon  was  on  the  meridian  of  the  places  whose  longitudes 
referred  to  Washington  are  respectively 


h.     in.        b.       m. 


h.       m. 


3  1-4-6  43-9=  -3  42-5=  -3*708 
3  1-4  +  6  34-3= +9  35-7  =+9-595. 


§71.] 


RELATED  PEOBLEMS 


139 


The  further  computation  is  then  as  follows : 

h.         m. 

Mean  Time  of  transit  over  Washington     -       5  58 -0 

Change  in  -3-708  h.  (-3-708  x  1-816  m.)    -0  6'7 

„   +9-595  h.  (  +  9-595  x  1-816  m.)    +0  17'4 

Mean  Time  of  transit  over  1st  point         -       5  51-3 

„      2nd    „    -         .6  15-4 

The  longitudes  of  these  two  points  referred  to  San  Francisco 
are  the  two  values  of  h  found  above.     We  therefore  have 
Mean  Time  of  moon-rise  at  San  Francisco : 

h.      m.          h.     m.  h.      m. 

1892,  June  1,  5  51-3-6  43'9  =  May  31,  23  7 -4 

=  June  1,  11  7'4  A.M. 

Mean  Time  of  moon-set  at  San  Francisco : 

h.      m.          h.        TD.  h,      TD. 

1892,  June  1,  6  15-4  +  6  34'3=June  1,  12  49-7 

=  June  2,    0  49'7  A.M. 

As  a  test  of  the  sufficiency  of  the  approximation,  we  now  com- 
pute the  moon's  declination,  with  the  times  which  we  have 
just  found  as  arguments.  The  result  is  : 

At  moon-rise,  8=  +  14°23'-6 
At  moon-set,  8=  +11  31  -3 

The  agreement  of  these  values  with  the  values  we  started 
-out  with,  shews  that  a  further  approximation  is  unnecessary. 

PROBLEM  XI.  To  find  the  sidereal  time  required 
for  the  semi-diameter  of  the  sun  or  moon  to  pass 
the  meridian. 

This  problem  arises  when,  from  the  observed 
R.A.  of  the  sun's  or  the  moon's  limb  at  the  moment 
of  transit,  it  is  required  to  find  the  R.A.  of  centre 
at  the  moment  of  transit  of  the  centre. 

Let  P  be  the  pole,  0  the  centre  of  the  moon,  L 
the  point  of  its  limb  tangent  to  PL,  Aoc  the  angle 
P,  S"  the  angular  semi-diameter  OL  expressed  in 
.seconds  of  arc. 


FIG.  13. 


140          MEASURE   OF  TIME   AND  RELATED  PROBLEMS      [§  71. 

Then,  (§  7,  Th.  ii.)  owing  to  the  smallness  of  the  arc  s  (16'), 

£"  =  sin  PO  .  Aa  =  cos  S  .  Arx. 
Hence,  Ao.  =  £"sec<$, 

or  when  reduced  to  time,     Aoc  =  Tlg  $"  sec  S. 

By  this  formulae  is  found  the  difference  of  R.A.  between  the 
centre  and  limb  at  any  moment.  But  what  we  want  is  the 
difference  of  R.A.  at  the  respective  moments  of  the  transits, 
which  is  different  owing  to  the  change  of  R.A.  during  the  time 
occupied  by  the  passage  of  the  semi-diameter.  To  find  the  time 
we  put 

OL,  the  R.A.  of  centre  at  transit. 

r,  the  sidereal  time  required  for  transit  of  semi-diameter. 

oc,'  the  change  of  R.A.  in  one  second  of  sidereal  time. 

(  =  change  in  1  m.  of  mean  tinie-i-60'17.  But  we  may  take 
60  as  the  divisor.) 

Then;  R.A.  of  centre  at  transit  of  limb  =  ot+a'T. 

R.A.  of  limb        „  „  =  oc  +  CL'T  +  AOL 

(Because  the  moment  is  the  same.) 
R.A.  of  limb  at  transit  =  OC  +  T. 
Hence,  by  equating  the  last  two  expressions 

Aa         S"  sec  8 


- 


(1) 


In  the  case  of  the  planets  OL  and  s  are  so  small  that  we  may 
use  the  formula 

8"  sec  8 


15 


(2) 


except  in  the  case  of  Venus  near  inferior  conjunction. 

Remark.  The  student  should  be  able  to  shew  that  (1)  will 
give  a  correct  result  for  every  place  by  using  the  geocentric 
values  of  S",  S,  and  a,  instead  of  their  apparent  values  as  seen  by 
the  observer. 


CHAPTER   VI. 

PARALLAX  AND   RELATED   SUBJECTS. 

Section  I.    Figure  and  Dimensions  of  the  Earth. 

72.  The  positions  of  the  heavenly  bodies,  as  found  from  astro- 
nomical tables,  and  given  in  ephemerides,  are  referred  to  the 
centre  of  the  earth ;  while  all  observations  upon  them  are  made 
on  its  surface.  Hence,  in  order  to  express  the  position  of  a  body 
referred  to  the  station  of  an  observer,  we  require  a  method  of 
reducing  its  coordinates  from  the  centre  of  the  earth  as  an  origin 
to  any  point  on  its  surface.  Such  a  reduction  requires  a 
knowledge  of  the  figure  and  dimensions  of  the  earth.  Strictly 
speaking,  what  we  should  know  in  order  to  make  the  reduction 
with  rigour  is  the  actual  figure  of  the  earth's  surface,  including 
all  the  inequalities  of  mountains  and  valleys,  because  all  points 
of  observations  are  situated  on  the  actual  surface.  The  figure 
of  the  latter  being  incapable  of  geometrical  definition,  the  ideal 
figure  used  in  geodesy  is  that  of  the  ocean  level,  and  is  called 
the  geoid.  The  surface  of  the  geoid  is,  at  any  point,  the  level  to 
which  the  water  of  the  ocean  would  flow  if  a  canal  or  tunnel 
were  cut  from  the  ocean  to  the  point. 

It  is  a  theorem  of  mechanics  that,  were  the  earth  homogeneous, 
the  geoid  would  be  an  oblate  ellipsoid  of  revolution.  In  reality, 
however,  the  heterogeneity  of  the  earth's  interior  and  the  attrac- 
tion of  mountains  is  such  that  the  surface  of  the  geoid  is  not 
rigorously  represented  by  any  definable  solid.  An  approximation 
sufficiently  near  for  most  geographical  and  astronomical  pur- 
poses is  obtained  by  considering  it  to  be  an  elliptic  spheroid  of 


142  PAEALLAX  AND   RELATED   SUBJECTS  [§  72. 

revolution,  affected  by  small  inequalities  which  are  to  be  deter- 
mined by  observation  in  each  region.  In  researches  relating  to 
parallax  the  inequalities  may,  in  all  ordinary  cases,  be  neglected,, 
and  the  geoid  considered  as  an  ellipsoid  of  revolution. 

Astronomical  observations  are  sometimes  made  at  considerable 
elevations  above  the  sea  level.  The  Lick  Observatory,  in 
California,  is  at  an  altitude  of  4400  feet.  This  altitude,  at  the 
mean  distance  of  the  moon,  would  subtend  an  angle  of  0"'7  ;  it 
should,  therefore,  be  taken  account  of  in  computing  the  parallax 
of  the  moon.  All  the  other  heavenly  bodies  are  so  distant  that 
the  elevation  of  the  observer  above  the  sea  level  may  be  left  out 
of  consideration. 

73.  Local  deviations. 

The  earth's  centre  being  invisible,  we  have  no  direct  way  of 
determining  its  direction  from  any  point  on  the  earth's  surface. 
The  only  line  of  reference  which  we  can  use  in  the  determination 
of  the  direction  of  a  heavenly  body  is  the  direction  of  gravity,  or 
that  of  the  plumb  line.  To  refer  observations  to  the  centre  of 
the  earth,  we  must  ascertain  the  figure  and  dimensions  of  the 
earth  from  geodetic  measurements  on  various  parts  of  its  surface,, 
combined  with  observations  of  the  force  of  gravity,  and  infer 
from  these  where  the  centre  is  located. 

In  doing  this,  an  element  of  uncertainty  is  introduced  by 
deviations  in  the  direction  of  the  plumb  line  due  -to  the  non- 
homogeneity  of  the  earth.  Since  the  attraction  varies  inversely 
as  the  square  of  the  distance,  and  is  exerted  by  every  part  of  the 
earth's  mass  according  to  the  law  of  gravitation,  those  portions 
in  the  neighbourhood  of  any  region  exert  a  preponderating 
influence  upon  the  resultant  direction  of  gravity.  Hence  if  the 
density  of  the  interior  is  greater  on  one  side  of  a  station  than  on 
the  opposite  side,  a  deviation  will  result.  In  mountainous- 
regions  deviations  are  observed  which  sometimes  amount  to  10",. 
or  20",  or  even  more.  Even  in  plains  far  distant  from  mountains 
such  inequalities  amounting  to  1"  more  or  less  are  the  general 
rule.  They  are  termed  local  deviations.  Since  the  surface  of 
the  geoid  is  everywhere  normal  to  the  direction  of  the  plumb 


§  74.]  LOCAL  DEVIATIONS  143- 

line,  these  deviations  shew  that  it  is  a  spheroid  with  numerous 
small  inequalities  all  over  its  surface. 

Since  the  astronomical  observations  of  altitude  are  referred 
to  the  direction  of  the  plumb  line,  it  follows  that  there  will 
be  corresponding  inequalities  in  the  celestial  and  terrestrial 
meridians  of  places  on  the  earth's  surface.  The  plane  of  the 
meridian  does  not,  as  a  general  rule,  pass  rigorously  through 
the  axis  of  the  earth.  It  must  be  defined  as  a  plane  containing 
the  vertical  line  and  parallel  to  the  axis. 

This  plane  defines  the  apparent  celestial  meridian  of  a  place 
in  the  following  way.  Imagine  that,  having  determined  a  north 
and  south  line  on  the  earth's  surface  by  the  above  condition, 
we  follow  it  a  short  distance  in  either  direction,  and  then  again 
determine  the  meridian.  We  shall  find  that  the  latter  will  not 
necessarily  be  a  continuation  of  the  first  meridian,  but  another 
line  making  a  minute  angle  with  it.  In  the  same  way,  the 
celestial  meridians  of  the  two  points  will  be  great  circles  inter- 
secting each  other  at  angles  which  we  may  regard  as  infini- 
tesimal. The  practical  meridian  found  by  starting  from  any 
point,  and  continually  travelling  in  the  apparent  north  and 
south  direction,  will  be  the  envelope  of  the  intersections  of  the 
system  of  meridian  planes  with  the  earth's  surface,  and  not 
rigorously  the  intersection  of  any  one  plane.  But  it  is  only  in 
refined  geodetic  work,  such  as  running  a  meridian  line,  that  the 
deviation  of  this  envelope  from  a  great  circle  is  of  importance. 
In  discussing  astronomical  observations  the  local  deviation  will  be 
unimportant  except,  possibly,  in  certain  observations  of  the  moon. 

74.  Geocentric  and  astronomical  latitude. 

The  inequalities  just  described,  combined  with  the  ellipticity 
of  the  earth,  lead  us  to  recognize  three  sorts  of  terrestrial 
latitude.  One  of  these — the  only  one  which  admits  of  being 
determined  by  direct  observation — is  the  angle  between  the 
plumb  line  and  the  plane  of  the  equator.  As  this  has  to  be  deter- 
mined by  astronomical  observation,  it  is  called  the  astronomical 
latitude. 

The  geocentric  latitude  of  a  point  on  the  earth's  surface  is  the 


144 


PAEALLAX  AND  RELATED  SUBJECTS 


angle  which  the  radius  vector  drawn  from  the  point  to  the 
earth's  centre  makes  with  the  plane  of  the  equator.  This 
latitude  is  that  which  has  to  be  used  in  computations  relating 
to  parallax.  It  does  not  admit  of  direct  determination,  but  has 
to  be  determined  by  correcting  the  astronomical  latitude  for  the 
difference  between  the  two  latitudes  as  inferred  from  geodetic 
measures  generally. 

A  third  latitude,  known  as  geographic,  is  sometimes  used. 
It  may  be  defined  as  the  astronomical  latitude  corrected  for  local 
deviation  of  the  plumb  line,  or  as  the  angle  made  with  the  plane 
of  the  equator  by  a  normal  to  the  surface  of  an  imaginary  geoid 
formed  by  smoothing  off  the  inequalities  of  the  actual  geoid 
so  as  to  reduce  it  to  an  ellipsoid  of  revolution.  As  this  latitude 
is  required  only  in  map-making,  where  great  precision  is  not 
necessary,  the  fact  that  it  does  not  admit  of  rigorous  deter- 
mination becomes  of  little  importance. 

For  our  present  purpose  the  problem  is  to  express  the  co- 
ordinates of  a  point  of  observation  when  referred  to  the  centre 
of  the  earth  as  an  origin,  in  terms  of  the  astronomical  or 

O          ' 

geographic  latitude. 

75.  Geocentric  coordinates  of  a  station  on  the  earth's  surface. 
Let  Fig.  14    represent  a  section  of  the  earth  through  the  axis, 
Y  being  the  north  pole.     The  earth  being  supposed  an  ellipsoid 


FIG.  14. 


of  revolution,  let  P  be  a  point  on  its  surface,  PZ  the  vertical 
determined  by  gravity,  normal  to  the  surface  of  the  geoid,  OPZ' 


§75.]  GEOCENTRIC  COORDINATES  145 

the  radius  vector  of  P  from  the  earth's  centre,  continued 
outwards. 

Then  Z  is  the  apparent  or  astronomical  zenith,  Z'  the 
geocentric  zenith. 

The  angle  ZPZ'  between  the  apparent  and  geocentric  zeniths 
is  called  the  angle  of  the  vertical. 

Omitting  local  deviation,  the  geocentric  zenith  is  on  the 
meridian  in  the  direction  from  the  apparent  zenith  toward  the 
celestial  equator.  Let  us  now  put  : 

0,  the  angle  XMP,  the  astronomical  latitude  of  P. 

</>',  the  angle  XOP,  its  geocentric  latitude. 

x,  y,  the  rectangular  coordinates,  OQ  and  QP,  of  P  referred 
to  the  principal  axes  OX  and  OF. 

p,  the  radius  vector  OP. 

a,  b,  the  major  and  minor  semi-axes,  OX  and  0  Y. 

6,  the  eccentricity  of  the  meridian  X  F. 

The  quantities  supposed  known  are  the  dimensions  and  form 
of  the  geoid,  expressed  by  a,  b,  and  e,  and  the  astronomical 
latitude  of  the  place,  found  by  direct  observation.  The  quan- 
tities required  for  parallax  are  p  and  <£'.  We  adopt  the  usual 
notation  and  formulae  of  analytic  geometry.  From  the  equation 
of  the  normal  it  follows  that  the  angle  which  ZP  makes  with 
the  major  axis  is  given  by  the  equation 


tan0  ......................  (1) 

From  the  equation  of  the  ellipse  we  have 


(2) 

from  which  we  are  to  determine  x  and  y  in  terms  of  <p. 
From  the  equation  (1)  we  derive  by  multiplication  by  C( 

b2x  sin  0  =  a2y  cos  0 

or  ¥x2sm-(f>  —  a4?/2eos20  =  0. 

From  this  equation  and  (2)  we  find 
„  a4  COB*  4 


a2  cos2    +     sn 


2_ 


146  PARALLAX  AND   BELATED   SUBJECTS  [§75, 

Introducing  the  eccentricity  by  the  substitution  62  =  a2(l—  e2)r 
we  have 


Thus,  introducing  p  and  0',  we  have 

a  cos  0       1 

x  =  pcos<j>  =     ,  *==== 

Vl-e2sm^l 

,     a(l-e2)sin0 
y  =  p  sin  0  =  —  V-= 

*  22 


To  find  the  angle  of  the  vertical,  0  —  0',  we  use 
sin  (0'  —  0)  =  cos  0  sin  0'  —  sin  0  cos  0'. 

Substituting  for  sin  </>'  and  cos  0'  their  values  derived  from  (3) 
and  noting  that  (3)  gives 


we  find 


,.      1  a-       e2  sin  20         1  e2  sin  20  /KX 

—  d>)  =  —  —  /==          y      =-  —  .  -y       —  .    ...(o) 

2      //l-62sin2        2  Vl-2e2- 


76.  Dimensions  and  compression  of  the  geoid. 

Instead  of  the  eccentricity  of  the  terrestrial  meridian  it  is 
common  to  use  its  compression  or  ellipticity.  By  this  is 
meant  the  fraction  by  which  the  ratio  of  the  semi-axes  differs 
from  unity.  Putting  c  for  this  quantity,  we  have 

a  —  b 


The  numerical  value  of  the  compression  is  still  somewhat 
uncertain  owing  to  the  small  extent  of  the  earth's  surface  over 
which  precise  geodetic  measures  have  been  extended.  Indeed, 
from  the  very  nature  of  the  case,  the  compression  must  be  a 
somewhat  indefinite  quantity,  there  being  no  one  spheroid  which 
we  can  exactly  define  as  fitting  the  surface  of  the  geoid  better 
than  any  other. 


§76.]    DIMENSIONS  AND  COMPEESSION   OF  THE   GEOID       147 

The  dimensions  of  the  geoid  as  determined  by  Bessel  many 
years  ago  are 

a  =  6  377397  metres  =  6  974532  yards, 
6  =  6  356  079  metres  =  6  951  218  yards ; 

Whence  C  =  29M5' 

e  =  0-0816967. 

These  numbers  have  been  generally  used  in  astronomy  and 
geodesy  for  the  greater  part  of  a  century.  During  that  interval 
geodetic  measures  have  been  greatly  extended.  A  general  de- 
termination made  by  Clarke  of  England  from  geodetic  measures 
is  a  =  6378249  metres, 

6  =  6356515  metres; 

whence  c  = 


293-5' 

e  =  0'08248. 

Clarke's  investigations  also  shew  that  the  actual  figure  could 
be  a  little  better  represented  by  an  ellipsoid  with  three  unequal 
axes,  the  equator  itself  having  a  slight  ellipticity.  It  is  probable, 
however,  that  this  apparent  ellipticity  of  the  equator  arises  from 
the  irregularities  with  which  the  actual  figure  of  the  earth  is 
affected. 

As  yet,  geodetic  measures  cover  so  small  a  fraction  of 
the  earth's  surface  that  an  accurate  determination  of  the  com- 
pression cannot  be  derived  from  them.  Measures  of  the  force  of 
gravity,  as  given  by  the  length  of  the  seconds'  pendulum,  are 
therefore  still  most  relied  upon  for  the  purpose  in  question.  It 
is,  therefore,  considered  by  the  best  authorities  that  Bessel's 
value  of  the  compression  is  nearer  the  truth  than  Clarke's. 
Helmert,  from  a  study  of  all  the  data,  has  recently  derived 
numbers  which  will  be  found  in  Appendix  I.,  and  which  may  be 
regarded  as  the  best  yet  reached. 


148 


PARALLAX   AND   RELATED   SUBJECTS 


[§77. 


Section  II.    Parallax  and  Semi-diameter. 

77.  The  word  parallax,  in  its  most  general  sense,  means  the 
difference  between  the  directions  of  an  object  as  seen  from  two 
different  points.  If  0  (Fig.  15)  be  the  object,  and  P  and  -Q  the 
points  of  observation,  the  parallax  is  the  difference  between  the 
directions  PO  and  QO.  Its  magnitude  is  measured  by  the  angle 
P'OQ'  =  POQ  between  the  lines  from  P  and  Q  to  0. 


FIG.  15. 


When  used  without  any  other  qualifying  adjective,  parallax 
commonly  means  the  difference  in  the  directions  of  a  heavenly 
body  as  seen  from  the  point  of  reference,  which  may  be  the 
centre  of  the  earth  or  of  the  sun,  and  from  some  point  of 
observation  on  the  surface  of  the  earth. 


FIG.  16. 

Parallax  in  altitude  is  the  difference  between  the  geocentric 
and  apparent  altitude  of  a  body.  If  in  Fig.  16,  P  is  the  body, 
Q  the  point  of  observation,  and  0  the  centre  of  the  earth,  the 
parallax  in  altitude  is  the  angle  OPQ. 

If  Q  is  so  situated  that  the  body  is  in  its  horizon,  say  at  J9, 
the  parallax  OBQ  is  called  the  horizontal  parallax. 


§77.]  PARALLAX   AND  SEMI-DIAMETER  149 

If  also  the  point  Q  is  on  the  earth's  equator,  so  that  OQ  is  the 
equatorial  radius  of  the  earth,  the  angle  OBQ  is  called  the 
equatorial  horizontal  parallax  of  the  body. 

It  will  be  seen  that  the  horizontal  parallax  is  equal  to  the 
semi-diameter  of  the  earth  as  seen  from  the  body. 

By  annual  parallax  is  meant  the  parallax  when  the  point 
of  reference  is  the  sun  and  that  of  observation  the  earth. 

By  parallax  in  any  coordinate  is  meant  the  difference  between 
the  values  of  that  coordinate  when  referred  to  the  centre  of  the 
earth  as  the  origin,  and  when  referred  to  a  point  on  its  surface. 
Thus  we  have  parallax  in  R.A.,  in  Dec.,  in  Latitude,  in  Longitude, 
etc. 

The  horizontal  parallax  is  connected  with  the  radius  of  the 
earth,  OQ,  and  the  geocentric  distance,  OB  of  the  body,  by  a 
simple  relation.  If  we  put 

p,  the  radius  of  the  earth  at  the  point  of  observation ; 

r,  the  geocentric  distance  OB ; 

TT/O  the  horizontal  parallax  of  B, 

TTV  the  equatorial  horizontal  parallax, 

we  have  r  —  p  sin  TO*. 

Hence  sin  TT/,=  ~ (1 ) 


r 


To  express  p  and  r  in  terms  of  the  same  unit  of  length,  we 
remark  that,  in  case  of  a  planet,  r  is  expressed  in  terms  of  the 
earth's  mean  distance  from  the  sun,  while  p  is  commonly 
expressed  in  terms  of  the  equatorial  radius  of  the  geoid.  Hence, 
if  p  and  r  in  (1)  are  expressed  in  this  way,  their  quotient  in  (1) 
must  be  multiplied  by  the  ratio  of  the  two  units,  which  is  the 
sine  of  the  sun's  mean  equatorial  horizontal  parallax  =  TTQ.  We 
may  then  write  instead  of  (1) 


For  the  equatorial  horizontal  parallax  of  the  planet,  which  is 
given  in  the  ephemeris,  we  write  1  for  p  and  ^  for  irh  in  (1). 

In  the  astronomical  ephemerides  the  equatorial  horizontal 
parallaxes  of  the  principal  bodies  of  the  solar  system  are 


150  PARALLAX  AND  RELATED  SUBJECTS  [§  77. 
given.  They  are  connected  with  the  distance  of  the  body  by 
the  relation  rrin^a,  (2) 

a  being  the  equatorial  radius  of  the  geoid,  expressed  in  the 
same  unit  as  r. 

78.  Parallax  in  altitude. 

There  being  two  radii  vectors  of  the  body,  one  from  the 
observer  and  one  from  the  earth's  centre,  and  two  zeniths,  there 
are,  in  all,  four  altitudes  and  zenith  distances  to  be  distinguished. 
We  shall  term  a  Z.D.  measured  from  the  geocentric  zenith  Z' 
(Fig.  14)  a  reduced  Z.D.,  and  one  defined  by  the  radius  vector 
from  the  earth's  centre  a  geocentric  Z.D. 

The  effect  of  parallax  is  evidently  to  make  the  apparent 
greater  than  the  geocentric  Z.D.,  the  azimuth  when  referred  to 
that  zenith  Z'  being  unchanged.  When  a  body  is  on  the  meridian 
the  geocentric  and  apparent  zenith  lie  on  the  same  great  circle 
with  it,  and  the  parallax  has  the  same  effect  on  the  reduced 
and  the  apparent  Z.D.  But,  if  the  body  is  not  on  the  meridian, 
the  displacement  by  parallax  will  not  take  place  on  a  vertical 
circle,  and  both  the  altitude  and  azimuth  of  the  body  will  be 
changed  by  it.  The  rigorous  determination  of  the  parallax  in 
altitude  and  azimuth  requires  the  solution  of  a  spherical  quad- 
rangle of  which  the  vertices  are  the  two  zeniths,  and  the 
geocentric  and  apparent  positions  of  the  body.  The  cases  in 
which  this  solution  is  necessary  are  so  rare  that  they  need  not  be 
considered  here.  Parallax  in  altitude  is  commonly  required  only 
in  the  case  of  a  body  on  the  meridian. 

To  find  the  parallax  in  altitude  on  the  meridian,  we  put 

7rv  the  equatorial  horizontal  parallax. 

7ra,  the  parallax  in  altitude. 

v,  the  angle  of  the  vertical,  taken  positively  in  the  northern 
hemisphere. 

0,  the  apparent  zenith  distance  of  the  body,  positive  toward 
the  south. 

z,  the  reduced  Z.D. 
Then  z  —  z  —  v. 


$  79.]  PARALLAX  IN  ALTITUDE  151 

From  the  definitions  already  given  and  the  constructions  in 
Figs.  14  and  16,  we  have  the  following  relations  between  the 
parallaxes  and  the  geocentric  distance  r  of  the  heavenly  body : 

From  (2);  sin  7^  =  -, 

From  (1 ) ;  sin  TTA  =  -  =  -  sin  7rr 

if     (i 

In  the  triangle  OQP, 

angle  OQP  =  180°-z': 
angle  OPQ  =  7ra; 
side  OQ  —  p', 
sideOP  =  r; 

whence,  by  the  law  of  sines, 

p  _  sin  7ra  . 
r~  sinz'  ' 

sin  7Ta  =  sin  z'  sin  irh. 
Whence  we  have  for  the  parallax  in  altitude 

sin  7ra  =  -  sin  (2  —  v)  sin  7^ (3) 

d> 

This,  being  subtracted  from  the  apparent  Z.D.  gives  the 
geocentric  Z.D. 

In  the  computation  of  parallaxes  we  take  the  equatorial  radius 
of  the  earth  as  unity,  and  use  the  symbol  p  to  designate  the 
ratio  of  the  radius  vector  of  the  place  to  the  equatorial  radius. 

Thus  the  preceding  expression  becomes 

sin  7Ta  =  p  sin  ^  sin  (z  —  v) (4) 

In  the  case  of  all  heavenly  bodies  except  the  moon  we  may 
assume  sin  TT  to  be  identical  with  TT  itself. 

79.  Parallax  in  right  ascension  and  declination. 

To  determine  the  difference  between  the  right  ascension  or 
declination  of  a  body  as  seen  from  the  centre  and  from  the 
surface  of  the  earth,  we  first  express  the  positions  of  the  observer 


152  PARALLAX  AND  RELATED  SUBJECTS  [§79. 

and  of  the  body  in  rectangular  coordinates,  the  origin  being  at 
the  centre  of  the  earth  and  the  axes  of  reference  as  follows  : 

Z,  the  axis  of  rotation  of  the  earth,  positive  toward  the 
north. 

X,  the  equatorial  radius  of  the  earth  in  the  meridian  of  the 
observer.  This  axis  cuts  the  celestial  equator  on  the 
meridian. 

Y,  an  equatorial  radius  cutting  the  earth's  surface  90°  west 
from  the  axis  of  X.  This  axis  cuts  the  celestial  sphere 
in  the  west  point  of  the  horizon  of  the  place.  Its 
positive  direction  is  the  opposite  of  the  conventional 
one,  in  order  to  correspond  to  the  usual  measure  of 
the  hour-angle. 

Then  putting  h  for  the  west  hour  angle  of  the  body  and  r 
and  S  for  its  geocentric  distance  and  declination,  we  have  the 
following  expression  for  its  rectangular  coordinates  : 

X  =  T  COS  S  COS  li\ 

y  =  r  cos  6  sin  h  V  .........................  (5) 

z  =  r  sin  £ 

From  the  definitions  of  the  coordinates  just  given  the  observer 
lies  in  the  plane  XZ.  His  coordinate  z  is  that  which,  in  treating 
the  figure  of  the  earth,  we  called  y.  Putting  (-,  ;/,  f  for  his  co- 
ordinates referred  to  the  present  system  of  axes,  we  have  : 


(6) 


The  last  quantities  are  determined  from  the  latitude  of  the 
observer,  as  already  shewn.  Putting  x',  y',  z  for  the  coordinates 
of  the  body  relative  to  the  observer,  we  then  have  : 


x  =  x  — 


We  also  distinguish  the  distance  r,  and  hour-angle  h'  of  the 
body  as  affected  by  parallax,  by  accents.     Then  substituting  in 


§  80.]      PARALLAX   IN   ASCENSION   AND  DECLINATION          153 

the  last  equations  for  the  rectangular  coordinates  their  expres- 
sions in  (5)  and  (6),  we  have 

r  cos  $'  cos  h'  =  r  cos  S  cos  h  —  p  cos  <^\ 

r  cos  S'  sin  li  =  r  cos  8  sin  h  - (7) 

r  sin  S'  =  r  sin  (5  —  p  sin  </>' 

The  geocentric  coordinates,  r,  S,  and  h,  being  given,  we  could 
from  these  equations  compute  r',  $',  and  A',  the  corresponding 
coordinates  relative  to  the  observer.  But  it  will  be  easier  to 
compute  the  parallax  in  R.A.  (or  hour-angle)  and  Dec.,  or  the 
values  of  h  —  h'  and  <$'  —  8.  The  problem  may  have  either  of  the 
following  two  forms : 

1.  Given,  the  geocentric  coordinates;  to  find  the  apparent  ones. 

2.  Given,  the  apparent  coordinates;  to  find  the  geocentric  ones. 

We  treat  the  problem  in  the  first  of  these  forms.  There  are 
also  two  methods  of  solution :  one  when  the  parallaxes  are  so 
small  that  their  second  powers  may  be  neglected ;  the  other  when 
this  is  not  the  case.  The  first  of  these  is  the  case  for  all 
heavenly  bodies  except  the  moon.  For  the  latter  the  solution 
should  be  rigorous. 

80.  Transformed  expression  for  the  parallax. 

We  transform  the  first  two  of  equations  (7)  as  follows. 
Multiplying  the  first  by  cosh,  the  second  by  sin  h,  and  adding 
the  products,  we  have  the  first  of  the  following  equations; 
multiplying  the  first  by  sin  h,  and  the  second  by  cos  h,  and 
taking  the  difference  of  the  products  we  have  the  second. 

r  cos  ff  cos  (h'  —  h)  =  r  cos  S  —  p  cos  </>'  cos  h\  /« \ 

r'  cos  ft  sin  (Ji  —h)  =  p  cos  (j>  sin  h  ) 

We  shall  now  put 

Aa  =  oL—oi  =  h  —  h',  the  parallax  in  R.A. 

A<5  =  $  —  S,  the  parallax  in  Dec. 

/y* 

f—  — ,  the  ratio  of  the  distance  of  the  body  from  the  observer 

to  that  from  the   earth's  centre.    /  is  a  little  less  than  unity, 
being  always  contained  between  the  limits  0'98  and  1. 


154  PARALLAX  AND  RELATED   SUBJECTS  [§  80. 

Divide  the  equations  (6)  and  (7)  or  (8)  by  r,  and  note  that, 
taking  the  earth's  equatorial  radius  as  unity,  we  have 

1 

r  =  —. . 

sin  TT-L 

-TTp  the  equatorial  horizontal  parallax,  is  taken  as  given,  and  is 
found  in  the  Ephemeris. 
Putting  for  brevity 


f  '  =  f  sin  TTj  =  p  sin  0'  sin  ir 
the  equations  (8)  and  (7)3  become 

/cos  ($'  sin  Aoc  =  —  £'  sin  /t         "j 

/  cos  <$'  cos  Aoc  =  cos  S  —  £'  cos  h  I  .................  (10) 

/sin<$'  =  sin<S-f 

The  quotient  of  the  first  two  equations  gives 

—  £'  sin  h 

tan  Aoc  =  --  ^—  r/  --  r  . 
cos  S  —  £  cos  h 

If  we  compute 

=     sec8,  ...............................  (11) 

this  equation  becomes 

(12) 


which  is  easily  computed  by  a  table  of  addition  and  subtraction 
logarithms,  especially  that  of  Zech.  It  may  be  yet  easier  to 
use  the  principal  table  of  Appendix  IV.,  the  form  (12)  being 
identical  with  that  for  the  precession  of  a  star  in  RA.  when  we 
replace  h  by  «,  and  assign  a  suitable  value  to  p.  For  this 
purpose  we  compute 

ps  =  [4T38  334]  p  cos  $'  sin  TTI  sec  3, 
enter  the  table  with  Arg.  ps  cos  h,  and  take  out  K. 

Then  Atoc  =  Kps  sin  h, 

Aoc  =  A«oc  —  red.  from  tangent  to  arc. 

In  the  case  of  the  declination  we  may,  instead  of  computing 
the  parallax,  compute  $  directly  from  the  equations  (10).  The 
quotient  (10)3cos  Aoc-:-(10)2  is 

.,     (sin  S  —  f  Ocos  Aoc 
tan<5  =  --  ^~  --  r- 
cos  3  —     cos  h 


I  81.]   TRANSFORMED  EXPRESSION  FOR  THE  PARALLAX      155 

This  direct  computation  of  &  requires  fewer  separate  quantities 
than  that  of  the  parallax;  but  7-figure  logarithms  will  be 
required  to  assure  the  result  being  correct  to  0"'l  whenever 
<$  >  10°.  As  5-place  logarithms  only  are  required  for  the 
parallax,  it  will  generally  be  easier  to  compute  the  latter. 

To  derive  the  formulae  for  the  parallax  in  Dec.  the  simplest 
formulae  for  computation  are  derived  by  multiplying  the  first 
two  equations  (30)  by  sinJAoc  and  cos  J  Aoc  respectively,  and 
adding.  We  thus  derive  the  second  of  the  following  equations, 
the  first  being  (10)3. 


where  we  write  for  brevity, 

a-  =  £'  cos  (h  —  J  Aoc)  sec  \  Aoc  ...................  (15) 

By  forming 

(14\  x  cos  <5-(14)2  X  sin  S  and  (14)x  x  sin  <S-f(14)2  X  cos  <J, 
and  adding,  we  have 

/sin  A($  =  <rsin(5  —  f'cos£       \  ,^ 

/cos  A3  =  1  -  f  '  sin  S  -  a-  cos  8)' 

To  facilitate  the  logarithmic  computation  of  these  equations 
compute  g  and  G  from 


we  shall  then  have 

tenA^-g^gr*)    .....................  (18) 

l-#cos(£—  S) 

a  form  similar  to  (12). 

81.  Mean  parallax  of  the  moon. 

The  moon  is  so  much  nearer  to  us  than  any  other  body  of  the 
solar  system  that  its  parallax  rests  upon  a  different  basis  from 
that  of  the  planets.  The  mean  value  of  its  parallax  is  called 
its  constant  of  parallax.  The  ratio  of  the  actual  parallax  at 
any  moment  to  this  constant  is  determined  from  theory  with  all 
the  precision  necessary  in  any  case  whatever.  But  the  actual 


156  PARALLAX  AND  RELATED   SUBJECTS  [§  81. 

value  of  the  parallax  as  tabulated  may  require  to  be  increased 
or  diminished  in  an  appreciable  ratio  on  account  of  possible 
error  in  the  constant. 

The  constant  in  question  has  been  determined  by  observations 
from  different  points  on  the  earth's  surface,  especially  at  the 
observatories  of  Greenwich  and  the  Cape  of  Good  Hope.  It  is 
also  determined  by  the  theory  of  gravitation,  the  problem  being 
at  what  distance  the  moon  should  be  placed  in  order  that  it  may 
revolve  around  the  earth  in  its  observed  time  of  revolution, 
allowance  being  made  for  the  disturbing  action  of  the  sun.  In 
this  form  the  problem  is  the  original  one  attacked  by  Sir  Isaac 
Newton  when  he  inquired  whether  the  moon  would  be  held  in 
her  orbit  at  the  observed  distance  by  the  gravitation  of  the 
earth,  the  latter  diminishing  as  the  square  of  the  distance.  The 
theoretical  method  now  affords  the  most  accurate  measure  of 
determining  the  distance  and,  therefore,  the  parallax  of  the 
moon.  The  best  result  of  theory  yet  obtainable  is  : 

Constant  of  equatorial  parallax  =  3422"*63. 

As  it  is  the  sine  of  the  parallax  which  enters  into  the  formulae, 
while   arc   is   most   conveniently  used  in  the   expression,  it  is 
common  to  use  the  sine  of  the  constant  instead  of  the  constant 
itself,  this  sine  being  reduced  to  seconds.     We  then  have 
Sine  of  constant  of  parallax  =  3422"'47. 

The  actual  sine  is  found  by  dividing  this  expression  by 
206  264"'8,  the  number  of  seconds  in  the  radius  unit,  or  multi- 
plying by  sin  1". 

In  Hansen's  tables  of  the  moon,  which  have  been  most  widely 
used  during  the  past  forty  years  in  the  computations  of  the 
ephemeris,  the  adopted  value  of  the  constant  is : 
Constant  of  sin  7r  =  3422"'07. 

If,  therefore,  the  best  value  of  the  parallax  is  required,  the  value 
from  the  ephemeris  should  be  increased  by  multiplying  it  by 
the  factor  1 '000  11 8.  Instead  of  multiplying  by  this  factor  the 
mean  value  of  the  correction,  -f-0"'40,  may  be  added  to  all  the 
values  of  the  equatorial  horizontal  parallax  in  the  Ephemerides 
without  an  error  exceeding  +  0"'03. 


§  83.]          PARALLAXES  OF  THE   SUN   AND  PLANETS  157 

82.  Parallaxes  of  the  sun  and  planets. 

As  the  parallax  of  the  nearest  planet,  Venus,  rarely  exceeds 
30",  quantities  of  the  second  order  as  to  the  parallax  of  the 
sun  and  planets  may  always  be  dropped,  thus  greatly  simplifying 
the  computation.     Putting 

TT/',  the  equatorial  horizontal  parallax  expressed  in  seconds 

of  arc, 
the  equations  (9)  will  become 

g'  =  p  cos  0V/'  sin  V, 
f  '  =  p  sin  0V/'  sin  1". 

Substituting  in  (12)  Aa"sin  1"  for  tan  Aoc  and  dropping  quan- 
tities in  p2,  we  have,  instead  of  (12), 

Aa"  =  —  pcos  0'  sin  h  sec  6V/'  ......  .............  (18) 

for  the  parallax  in  R.A.  expressed  in  seconds  ot  arc. 

With  the  same  abbreviation,  the  computation  of  the  parallax 
in  Dec.  takes  the  form 

g  sin  G  =  p  sin  0V 
ycos  G  =  p  cos  0  V"  cos  h 
A<T=-#sin(£-<$) 
which  will  be  the  parallax  in  Dec.  expressed  in  seconds  of  arc. 

83.  Semi-diameters  of  the  moon  and  planets. 

No  observations  have  yet  shewn  any  deviation  of  the  apparent 
disc  of  the  moon  from  the  circular  form,  local  irregularities  of 
the  surface  excepted.  The  figure  of  our  satellite  is,  therefore, 
treated  as  spherical.  The  linear  radius,  RM  is  commonly 
expressed  by  its  ratio  to  the  equatorial  radius  of  the  earth,  RE, 
and  is  called  k.  This  quantity  cannot  be  measured  directly,  but 
is  derived  from  the  observed  angular  semi-diameter  of  the  moon, 
combined  with  the  parallax,  taken  as  known.  Since  the  moon's 
parallax  is  the  earth's  semi-diameter  seen  from  the  moon,  it 
follows  that  if  we  put  $0,  the  moon's  angular  semi-diameter  at 
the  distance  corresponding  to  the  constant  of  parallax,  we  shall 
have,  for  the  ratio  of  the  radii  of  the  earth  and  moon, 


sn 


358  PARALLAX  AND  RELATED   SUBJECTS  [§  83. 

From  very  comprehensive  recent  discussions  of  occupations 
of  stars  by  the  moon,  made  by  Struve,  Peters,  and  Batter  mami, 
it  is  inferred  that  the  best  value  of  the  moon's  semi-diameter  at 
the  distance  corresponding  to  the  co'nstant  of  parallax  is 

S0  =  932"-57. 
The  corresponding  value  of  k  is 


Using  sin  ^  =  3422'47",  this  gives 

k  =  0-272  483. 

If  Hansen's  constant  of  parallax  is  used,  the  result  is 

&  =  0-272  516. 

In  the  case  of  a  planet,  if  we  put  r  for  its  linear  radius  at  any 
point  on  the  edge  of  its  apparent  disc,  as  seen  from  the  earth,  its 
apparent  angular  radius  to  that  point  is  found  by  the  same 
geometric  construction  as  the  horizontal  parallax  (§  77).  Putting 
s  for  the  angular  semi-diameter,  the  value  of  s  as  seen  by  an 
observer  at  the  distance  7*  from  the  centre  of  the  planet  is  given 
by  the  equation 

R 


sin  *  =  — 


.(20) 


The  figures  of  all  the  planets  may  be  treated  as  ellipsoids  of 
revolution,  of  which  the  eccentricity  vanishes  if  the  figure  is 
spherical.  Let  us  put  Ra  and  Rb,  the  semi-axes  of  the  ellipsoid. 
The  linear  radius  of  the  planet  at  latitude  0  will  then  be  given 
with  all  necessary  precision  by  the  equation 

JS  =  fia(l-c-sin20), (21) 

c  being  the  compression 

_Ra—Rb 

~Ka 

The  apparent  disc  is  a  spherical  ellipse  of  which  the  major 
semi-axis  is  found  by  putting  Ra  for  R  in  (20).  To  find  the 


S  83.]    SEMI-DIAMETERS   OF  THE   MOON  AND   PLANETS        159 

minor  axis,  and  its  position-angle  with  respect  to  the  hour-circle 
passing  through  the  planet,  let  us  put 

A,  D,  the  R.A.  and  Dec.  of  the  pole  H  of  the  planet's  axis 

of  rotation, 
oc,   8,  those   of    the   planet   itself,    whose   position  on    the 

celestial  sphere  we  call  K. 
P,  the  celestial  pole  of  the  earth. 

In  the  spherical  triangle  PHK  will  then  be 

Side  HK=9Q°  +  <j> (22) 


Angle  PHK  =  position-angle  of  minor  axis  relative  to  the 
hour-circle  through  K. 

We  assume  as  given  the  pole  H,  in  terms  of  its  R.A.  and  Dec. 
If  given  in  ecliptical  coordinates,  these  are  to  be  converted  into 
equatorial  ones.  Then,  the  solution  of  the  triangle  PHK  so  as 
to  find  the  angle  H  and  the  side  HK  will  give  us  <£  and  the 
position-angle  of  the  axes  of  the  apparent  disc. 

Practically  3-figure  logarithms  will  suffice  in  the  solution  of 
the  triangle. 

The  astronomical  data  usually  given  for  determining  semi- 
diameters  of  the  planets  are  the  apparent  angular  semi-diameters 
S0  at  some  standard  distance  r0,  for  which  the  unit  of  distance 
or  the  mean  distance  of  the  sun  is  generally  taken.  Whatever 
the  value  of  r0,  we  have,  for  the  apparent  semi-diameter 

^sins0.  ............................  (23) 


In  the  case  of  the  planets,  s  is  so  small  that  we  may  always 
use  the  semi-diameters  themselves  expressed  in  seconds  of  arc, 
instead  of  sin  s. 


CHAPTER  VII. 


ABERRATION. 

84.  The  observed  and  commonly  accepted  law  of  displacement 
of  a  star  by  aberration  is  this : 

Let  S  be  the  position  of  a  star  at  the  moment  when  a  ray  of 
light  leaves  it,  and  E  that  of  the  earth  when  the  ray  reaches  it. 

Let  EE'  be  the  direction  in  which 
the  earth  is  moving  at  the  moment, 
and  v  its  velocity.  Also,  put  F  for 
the  velocity  of  light. 

Now  draw  SS'  parallel  to  EE',  and 
of  such  length  that 

SS':SE=v:V. 

Then  the  law  in  question  is  that  the 
star  S  will  be  seen  by  an  observer  on 
E  in  the  apparent  direction  ES'. 
Stated  in  a  general  form  the  law  is : 

The  apparent  position  of  an  object  seen  by  an  observer  in 
motion  is  displaced  from  the  true  position  in  which  it  would  be 
seen  if  the  observer  were,  at  rest  by  an  amount  equal  in  linear 
measure  to  the  observers  motion  at  constant  speed  during  the 
time  occupied  by  light  in  passing  from  the  object  to  the  observer. 
The  direction  of  the  displacement  is  that  of  the  observer's  motion 
at  the  moment  of  observation. 

To  express  the  law  in  algebraic  form,  put 

v,  the  linear  speed  of  the  observer's  motion. 

A$,  the  apparent  displacement  in  linear  measure. 

R,  the  distance  of  the  object. 


FIG.  17. 


§84.]  LAW  OF  ABEERATION  161 

The  time  occupied  by  the  light  in  passing  will  then  be  R  -r-  V, 
.and  for  the  apparent  linear  displacement  we  have 


The  angular  displacement  of  the  object,  represented  by  SJES'  is 
called  its  aberration;  and  its  effect  upon  the  value  of  any 
coordinate  is  called  the  aberration  in  that  coordinate.  To  find 
its  amount  for  any  coordinate  let  us  put 

X,  Y,  Z,  the  rectangular  coordinates  of  S  referred  to  any 
system  of  axes  having  its  origin  in  E  ; 

x'y  y',  0',  the  components  of  the  velocity  of  the  earth  resolved 

in  the  direction  of  these  axes  ; 

AJT,  AF,  l±Z,  the   coordinates  of  8'  relative  to  8,  so  that 
the  coordinates  of  8'  relative  to  E  are 

X  +  AX;    F+AF;    Z+&Z. 

To  express  AX,  AF,  LZ  in  terms  of  x,  y',  and  z',  let  us  put 
«.,  /3,  y,  the  angles  which  the  parallel  lines  EE'  and  88'  make 
with  the  coordinate  axes.  We  then  have 

x'AS 


/A    CT 

z'  —  v  cos  y,        t±Z=SS'  cos  y  =  —  — . 
Also  R  being  the  distance  ES  from  the  earth  to  the  star, 

AS  =  ^£. 
We  then  have,  by  the  preceding  equations, 


y- (1) 

z' 

N.S.A.  L 


162  ABERRATION  [§  85, 

85.  Reduction  to  spherical  coordinates. 

These  are  the  general  expressions  for  the  apparent  displacement 
of  a  star  by  aberration,  and  are  valid  for  any  rectangular  system 
of  axes.  As  the  position  of  a  star  is  always  expressed  by 
spherical  coordinates,  we  must  reduce  the  expressions  to  the 
corresponding  ones  in  such  coordinates. 

Putting 

L,  the  longitude  of  the  star  in  any  such  system ; 
B,  its  latitude ; 
R,  its  distance, 

we  have 

X  =  R  cos  B  cos  L  \ 

(2) 


The  displacement  by  aberration  is  so  slight,  about  20",  that  we 
may  derive  its  value  from  these  expressions  by  differentiation. 
We  thus  find,  as  in  deriving  formulae  (4)  of  §  48, 

r  AX ,         r  AF 
cos  B Ai>  =  —  sin  L  — p—  +  cos  L  — „-, 

AD          .     p        .AX      .         .     rAF ,          DA£ 
A.D  =  —  sin x>  cos  jt/  — p- — sm />  sin //  — ~- +  008.0—75- . 
-ft/  MM 

Substituting  for  AX,  AF,  AZ,  their  values  (1),  we  find  for  the 
effect  of  the  aberration  upon  the  spherical  coordinates  of  the 
star, 

/  / 

cos  BkL  =  —  T^sin  L+^. cos  L 

xf 
A#  =  —  sin  B  cos  L  -y— sin  B  sin  L 

The  next  step  is  to  substitute  for  x',  y'  and  z',  the  resolved 
components  of  the  motion  of  the  earth,  their  expressions  in 
terms  of  the  elements  of  the  earth's  orbit.  This  requires  the  use 
of  the  elementary  formulae  for  the  elliptic  motion  of  the  earth 
round  the  sun,  which  we  assume  to  be  given. 


§  85.]  REDUCTION  TO   SPHERICAL  COORDINATES  163 

We  shall  first  take  for  L  and  B  the  ecliptical  coordinates  of 
the  star.  Let  us  put 

X,  the  longitude  of  the  earth  in  its  orbit  ; 
r,  the  earth's  radius  vector  ; 
TT,  the  longitude  of  the  earth's  perihelion  ; 
e,  the  eccentricity  of  its  orbit  ; 
/,  its  true  anomaly  ; 

n,  its  mean  angular  velocity  around  the  sun  ; 
A',  its  actual  angular  velocity,  or  the  value  of  Dt\  ; 
r',  the  value  of  Dtr  ; 

x,  y,  0,  the  rectangular  coordinates  of  the  earth  referred  to 
the  sun. 

When  referred  to  the  ecliptic  the  motion  of  the  earth  in 
latitude  is  so  minute  that  it  may  be  left  out  of  the  problem. 
We  therefore  put 

x  =  r  cos  A, 
2/  =  rsin  A, 
0  =  0. 
Then  by  differentiation 

&'  =  r'cosA  —  r  A7  sin  A 


By  the  law  of  elliptic  motion  of  a  planet,  it  is  shown  that 
r,  r',  and  A'  are  given  by  the  equations  '^ 

a_l  +  e  cos(  A  —  TT) 
r~         cos2^ 

a2 
X/==V™COS0> 

,  _  aen  sin  (A  —  TT) 

COS0 

where  0  is  the  angle  of  eccentricity,  defined  by  the  equation 

sin  0  =  e, 


whence  cos  <>  =  \l  — 


164  ABEREATION  [§  85. 

Substituting   these  values  of   X'   and  r'  in  (4),  we  find,   by 
suitable  reductions, 


| 
J 


y'  =     ail  sec  </>(cos  X  +  e  cos  TT) 

Substituting  these  values  in  the  equation  (3),  and  putting  for 
brevity 


we  find 

cos  BkL  —  K  (sin  A  +  e  sin  TT)  sin  L 
-+•  K  (cos  A  +  6  cos  TT)  cos  L 
=  KCOs(\  —  L)  +  eKcos(7r  —  L) 
AB  =  K  sin  B  sin  (A  —  L)  4-  e/c  sin  B  sin  (TT  —  //) 

Studying  the  last  terms  of  these  equations,  it  will  be  seen 
that  they  are  independent  of  the  earth's  longitude,  and  functions 
of  the  elements  of  the  earth's  orbit  and  the  coordinates  of  the 
star.  The  variation  of  these  quantities  is  so  slow,  and  the 
factor  so  minute,  that,  unless  the  star  be  in  the  immediate 
neighbourhood  of  the  pole,  the  terms  in  question  may  be  re- 
garded as  constant  for  several  centuries.  They  may,  therefore, 
be  left  out  of  consideration  for  the  present,  being  included  in 
the  values  of  the  coordinates  of  the  star  as  determined  by 
observation. 

In  the  usual  formulae  for  aberration  we  put 

0,  the  true  longitude  of  the  sun,  =A  — 180°. 

The  aberration  in  the  longitude  and  latitude  of  a  fixed  star 
may  therefore  be  expressed  in  the  form 

cos£AZ=  —  /ccos(0  — //)          \  ,gv 

Q-L)J 1 


'  =  —  AC  sin  B  sin  (  0 

86.  The  constant  of  aberration  and  related  constants. 

The  coefficient  K,  which  is  called  the  constant  of  aberration, 
demands  our  special  attention.  From  the  definitions  of  a  and  n 
it  follows  that  an  is  the  linear  velocity  which  the  earth  would 
have  if  it  moved  in  a  circular  orbit  of  radius  a.  As  the  earth 
actually  moves,  an  sec  <j>  is  the  half -sum  of  its  greatest  and  least 


§  86.]  ABERRATION   AND  RELATED  CONSTANTS  165 

velocities,  which  we  may  term  (though  not  with  strict  correct- 
ness) its  mean  velocity.  Thus  from  (6): 

The  constant  of  aberration  is  the  ratio  of  the  mean  velocity  of 
the  earth  in  its  orbit  to  the  velocity  of  light. 

This  mean  velocity  is  the  product  of  its  velocity  were  its 
orbit  circular  and  its  time  of  revolution  unchanged,  into  sec  0. 

There  are  two  ways  in  which  we  may  determine  the  constant 
of  aberration: 

(1)  By  observation  of   the   annual  change  in  the   R.A.   and 
Dec.  of  the  stars  produced  by  aberration.     By  the  most  refined 
measures  yet  made  the  constant  is  found  to  be  20"'52,  with  an 
uncertainty  of  2  or  3  hundredths  of  a  second. 

(2)  Supposing  the  dimensions  of  the  earth's  orbit  to  be  known, 
we   may   compute   the   velocity  of   the   earth.     We   have   also 
determined,  by  actual  measurement,  the  velocity  of  light.     Thus 
the  ratio  of  the  two  velocities  may  be  computed.     Let  us  put 

TTQ,  the  mean  equatorial  horizontal  parallax  of  the  sun ; 
p,  the  earth's  equatorial  radius. 

To  compute  the  mean  velocity  of  the  earth  in  its  orbit, 
retaining  TTO  as  an  unknown  quantity,  we  have  the  data : 

p     =  6378-2  Idl. 
~~  sin  TTo  ~~      sin  TTO 
Taking  one  second  as  our  unit  of  time,  we  have: 

Sidereal  year  =  365  d.  6  h.  9  m.  9  s.  =  31  558  149  s., 
circumf.        6'283 19 


'aid.  year     31  558  149' 
log  sec  0  =  0-000  061, 

[7103  83]     [2-418  25] 
L  J 


Sin  7To  7T 

Here,  the  number  in  brackets  is  the  logarithm  of  the  number  to 
be  used;  and  TT"  means  TTO  expressed  in  seconds  of  arc.  The 
second  fraction  is  derived  from  the  first  by  multiplying  its 
terms  by  the  number  of  seconds  in  radius  (206  265"). 

The  measurement  of  the  velocity  of  light  gives  the  result 
F=  299  860  kilometres  per  second. 


166                                              ABERRATION                                          [§  86. 
Hence  -  ..I«5«»*J,   (9) 

7T 

or,  if  we  express  K  in  seconds,  by  multiplication  by  206  265, 

[2-25576]     180-20 

*"** v — T, • 

7T  7T 

We  thus  have,  between  K"  and  ir"  the  fundamental  relation 

,c"X7r"  =  180-20 (10) 

We  have  retained  IT"  as  an  unknown  quantity,  because  it  is 
very  difficult  to  determine,  whereas  the  number  180*20  is 
probably  correct  within  3  or  4  units  of  its  second  place  of 
decimals. 

It  follows  that,  of  the  constant  of  aberration  and  the  solar 
parallax,  we  can  determine  the  one  when  we  know  the  other. 
They  can  be  determined  by  observation  with  perhaps  equal 
absolute  accuracy,  but  as  K"  is  more  than  twice  as  great  as  TT", 
this  implies  that  it  is  determined  with  greater  relative  accuracy. 
The  solar  parallax  can,  therefore,  be  determined  from  /c  with 
more  accuracy  than  in  any  other  one  way,  if  we  admit  the 
completeness  of  the  fundamental  theory  of  aberration. 

87.  Aberration  in  right  ascension  and  declination. 

This  may  be  determined  by  referring  the  position  of  the  star 
and  the  motion  of  the  earth  to  equatorial  coordinates,  which  are 
those  most  used  in  computations  relating  to  the  fixed  stars.  Let 
us  put 

xv  yv  zv  the  heliocentric  coordinates  of  the  earth  referred  to 
the  equatorial  system. 

x",  y",  z",  the  corresponding  velocities. 

The  transformation  from  the  ecliptic  system  to  the  equatorial 
system  is  found  by  writing  in  the  equations  (7)  or  (8)  of  §  49, 
x",  y",  z",  for  x,  y,  z, 
x',  y',  z,  for  Z,  F,  Z, 

which  gives  x  —  x', 

y"  =  y'  cos  e  —  z'  sin  e, 
z"  =  y'  sin  e  +  z'  cos  e. 


§87.]    ABERRATION   IN  ASCENSION  AND   DECLINATION       167 

We  thus  find  from  (5),  substituting  O  -f 180°  for  X,  and  taking 
IT  to  represent  the  longitude  of  the  solar  perigee, 

TT  =  281°  13'  in  1900; 

x"  =     an  sec  0  (sin  0  +  e  sin  TT)        ] 

y"  =  —  an  sec  <£  cose  (cos  O  +ecos7r)  V (11) 

0"  =  —  an  sec  0  sin  e  (cos  ©  +  e  cos  TT)  J 

If  we  also  put  Xv  Fp  Zv  the  rectangular  coordinates  of  the 
star  referred  to  the  equatorial  system,  the  equations  (1)  give  for 
its  displacement 

a     x" 

=  -    =  AC  (sin  0  +  e  sin  TT), 


.(12) 


— rT1  =  17  =  —  K  COS  €  (COS  O  -f  6  COS  TT), 
.£1  r 

A^,      0" 

—jo1  =  -y.  =  —  AC  sin  e  (cos  Q  +e  cos  TT). 

For  the  reason  already  mentioned  we  may  leave  out  of  account 
the  constant  terms  ?AC  cos  TT  and  eic  sin  TT,  and  write 

AX, 

— n±  =  AC  sin  0 

AF, 

— nr  =  —  AC  COS  e  COS  0 

— rr-  =  —  AC  sin  e  cos  0 

The  effect  on  the  R.A.  and  Dec.  of  the  star,  when  quantities  of 
an  order  higher  than  the  first  are  dropped,  is  found  by  putting 
in  (4a)  of  §  48,  A,  Xv  Ylt  Zv  and  R  for  d,  x,  y,  z,  and  r. 

cos  S Aoc  =  —  AC  cos  e  cos  0  cos  a  —  AC  sin  0  sin  a  "| 

AcS  =  AC  cos  e  cos  0  sin  ^  sin  a  — AC  sin  0  sin^cosocr ' 

—  AC  sine  cos  O  cos<?J 

The  form  in  which  these  equations  are  used  in  practice  will 
be  shown  in  the  chapter  on  the  reduction  of  places  of  the  fixed 
stars. 


168  ABERRATION.  [§  88. 

88.  Diurnal  aberration. 

In  the  preceding  theory  the  motion  taken  into  account  has 
been  only  that  of  the  centre  of  the  earth  around  the  sun.  But, 
in  consequence  of  the  earth's  rotation  on  its  axis,  the  observer  is 
continually  carried  toward  the  east  point  of  his  horizon  with  a 
speed,  in  metres  per  second, 

s0/,  ...........................  (14) 


<j>  being  his  geocentric  latitude,  and  p  the  radius  of  the  earth 
at  his  station  in  terms  of  the  equatorial  radius.  The  speed 
of  rotation  of  a  point  on  the  earth's  surface  at  the  equator 
is  464. 

The  effect  of  this  motion  is  to  produce  a  universal  displace- 
ment of  the  apparent  positions  of  all  the  bodies  in  the  heavens 
toward  the  east  point  of  the  horizon,  on  great  circles  passing 
through  this  point,  expressed  by 


0  being  the  distance  of  the  body  from  the  east  point.     Putting 
for  v  and  V  their  numerical  values  and  reducing  to  seconds, 

s  =  0"-319/>cos0'sin0  .........................  (15) 

To  find  the  effect  of  the  displacement  upon  the  R.A.  and 
Dec.  of  the  body,  consider  the  spherical  triangle  PES  formed 
by  the  pole  P,  the  east  point  of  the  horizon,  E,  and  the  body  £. 
Then  0  is  the  side  SE  ;  and  if  we  put 

5,  the  angle  at  S  ; 
h,  the  hour-angle  of  the  body, 
the  aberration  in  R.A.  and  Dec.  will  be  : 


We  have,  in  the  triangle, 

sin  0  sin  q  =  cos  h 

sin  0  cos  q  =  sin  3  sin  h. 


§  89.]  DIURNAL   ABERRATION  169 

Substituting  in  (16)  the  value  of  s  from  (15),  we  find 
Aoc=  0"'319  /o  cos  0' cos /t  sec  (51 
A£  =  0"-319  p  cos  $'  sin  S  sin  fc]  ' 

In  using  this  expression,  we  may  put  p  =  I  and  0'  =  0. 

In  reducing"  meridian  observations  due  correction  is  made  for 

o 

the  effect  of  diurnal  aberration.  Generally,  however,  it  is  ignored 
in  practical  astronomy,  because  it  affects  all  bodies  in  the  same 
region  by  the  same  amount :  and  that  amount  being  very  minute 
is  seldom  of  practical  importance.  It  should,  however,  be  taken 
into  account  in  all  investigations  involving  the  relative  positions 
of  widely  separated  bodies. 

89.  Aberration  when  the  body  observed  is  itself  in  motion. 

The  preceding  theory  is  based  on  the  relations  of  an  observer 
in  motion  to  a  ray  of  light  which  has  emanated  from  a  heavenly 
body,  the  possible  motions  of  that  body  being  left  out  of  con- 
sideration. Since  the  course  of  the  ray  depends  wholly  upon 
the  position  of  the  emitting  body  at  the  moment  when  the  ray 
left  it,  and  is  independent  of  the  position  of  that  body  at  any 
other  moment,  the  theory  already  developed  is  complete  for  the 
position  of  the  body  at  the  moment  in  question.  That  is  to  say, 
when  the  correction  for  aberration  is  applied  to  the  apparent 
position  of  a  body,  the  result  will  not  be  the  position  of  the  body 
at  the  moment  T  of  observation,  but  at  the  moment  T—r,  T 
being  the  time  required  for  light  to  come  from  the  body  to  the 
observer.  If,  therefore,  the  actual  direction  of  the  body  is 
required  for  the  time  T  of  observation,  its  motion  during  the 
interval  T  must  be  determined  and  added. 

The  general  theory  sets  no  limitations  upon  the  motion  of 
the  body  during  the  interval  occupied  by  the  passage  of  the 
light  to  our  system.  A  double  star  revolving  in  an  orbit  may 
make  several  revolutions  during  this  interval.  In  stellar  as- 
tronomy generally  no  account  is  taken  of  these  possible  motions 
or  changes.  The  fact  that  the  distance  of  the  stars,  and  there- 
fore the  time  T,  is  not  known  with  precision,  prevents  any 
accurate  determination  of  the  motion  during  this  interval,  and 
at  the  same  time  renders  it  unimportant.  All  our  statements 


170  ABEREATION  [§  89. 

respecting  what  is  going  on  among  the  stars  at  a  stated  time  T 
really  refer  to  phenomena  which  occurred  at  a  time  T—T,  T 
being  an  unknown  number  of  years,  which  we  regard  as 
constant  for  any  one  star  or  system,  and  which  is  left  out  of 
consideration. 

When  the  forces  which  may  possibly  be  at  play  among  the 
stars  admit  of  more  complete  investigation  than  they  now  do,  it 
may  be  that  the  variations  during  the  interval  T  will  enter  as 
important  elements  into  the  problem.  It  is  interesting,  if  not 
essential,  to  remark  that,  from  the  best  estimate  that  can  be 
made  of  the  distance  of  the  star  1830  Groonbridge,  its  actual 
direction  is  about  3'  ahead  of  its  observed  and  adopted  position, 
in  the  direction  of  its  proper  motion. 

90.  Case  of  rectilinear  and  uniform  motion. 

When  the  motion  of  the  observed  body  is  rectilinear  and 
uniform  during  the  time  T,  it  can  be  shown  that  its  displacement 
by  aberration  depends  solely  upon  the  relative  motions  of  the 
observer  and  the  body,  irrespective  of  the  absolute  motions  of 
either.  To  show  this,  let  us  put 

X',  Y,  Z',  the  components  of  the  speed  of  the  body  in  the 
direction  of  the  three  coordinate  axes. 

X0,  YQ,  Z0,  its  coordinates  at  the  moment  T—  r  when  the  light 
left  it  by  which  it  was  observed  at  the  time  T.  Its  actual  co- 
ordinates at  this  time  are  found  by  adding  to  X0,  Y0,  Z0,  the 
motion  during  the  time  T,  and,  therefore,  are 


7=F+r 


Its  apparent  coordinates  are  expressed  by  adding  to  XQ,  F0, 
and  Z0  the  displacements  given  by  the  equations  (1),  in  which 
wehave  R=VT  ...............................  (18) 

These  coordinates  are  therefore 


§91.]  ABERRATION  OF  THE   PLANETS  171 

The  differences  between  the  true  and  apparent  coordinates  at 
the  epoch  T  now  become 

Xap-X=(x'-X')T  \ 

Yap-Y=(y'-Y)T    I   (19) 

Zap-Z=(z'-Z')T    } 
which  depend  only  on  the  differences 

x'—X',  etc. 

The  last  numbers  of  these  equations  express  the  change  in 
the  coordinates  of  the  body  observed,  relative  to  the  moving 
earth  as  an  origin,  during  the  period  occupied  by  light  in  passing 
from  the  body  to  the  earth.  The  total  displacement  by  aberration 
is  therefore,  in  the  case  supposed,  equal  to  this  change.  If, 
instead  of  rectangular  coordinates,  we  use  R.A.  and  Dec.,  the 
expression  for  the  aberration  in  these  coordinates  will  be 

Ab.  inRA..=  — rAa  \  (2Q) 

Ab.  in  Dec.  -  -  TDtS  J ' 

To  find  T,  the  distance  R  of  the  body  must  be  known.  If  we 
express  R  and  V  in  terms  of  the  semi-major  axis  of  the  earth's 
orbit  as  unity,  we  shall  have  from  (18) 

R 

T=V> 

which  may  be  substituted  for  r  in  the  above  expression. 

91.  Aberration  of  the  planets. 

In  the  astronomical  ephemerides  there  are  two  systems  of 
-dealing  with  aberration  of  the  planets.  One  consists  in  giving 
the  apparent  coordinates  of  the  planets  at  the  epochs  of  the 
ephemeris,  commonly  mean  noon,  as  affected  by  the  aberration. 
These  apparent  coordinates  are  found  by  applying  the  corrections 
(20)  to  the  true  coordinates. 

As,  for  theoretical  purposes,  it  may  sometimes  be  desirable  to 
have  the  actual  position  of  the  planet  at  the  assigned  time,  and 
as,  in  the  case  of  newly  discovered  objects,  the  distance  may  be 
unknown,  the  method  is  sometimes  adopted  of  giving  at  the 


172  ABEERATION  [§  91. 

stated  epochs  the  actual  position  of  the  planet.  Then,  if  the 
distance  is  known,  the  value  of  r  can  be  computed.  If  from 
such  an  ephemeris  apparent  coordinates  of  a  planet  are  required 
at  a  given  time  T,  we  subtract  r  from  T  and  interpolate 
the  true  coordinates  to  the  moment  T—  T.  This  gives  us  the 
apparent  position  of  the  planet  affected  by  aberration  at  the 
time  T. 


CHAPTER  VIII. 
ASTRONOMICAL  EEFRACTION. 

Section  I.    The  Atmosphere  as  a  Refracting  Medium. 

92.  Astronomical  refraction  is  the  refraction  of  a  ray  of  light 
by  the  atmosphere  as  it  is  passing  from  a  celestial  object  to  the 
eye  of  the  observer.  Its  measure  is  the  change  produced  by  it  in 
the  direction  of  the  ray.  The  total  amount  of  refraction  depends, 
not  only  upon  the  density  of  the  various  strata  of  air  through 
which  the  ray  passes,  but  also  on  the  direction  of  the  strata  of 
^qual  density  with  respect  to  the  vertical.  When  the  atmos- 
phere is  in  a  condition  of  equilibrium,  these  strata  are  horizontal. 
But,  owing  to  aerial  currents  and  other  causes,  this  is  not 
universally  the  case.  The  deviation  from  horizontality  is 
specially  marked  in  the  strata  which  separate  the  air  inside  an 
observing  room,  and  even  inside  the  tube  of  a  telescope,  from 
the  external  air.  The  refraction  due  to  this  cause  belongs  to 
the  subject  of  practical  and  instrumental  astronomy,  and  there- 
fore will  not  be  considered  in  the  present  chapter.  For  the 
most  part  the  astronomer  is  under  the  necessity  of  neglecting 
all  irregularities  in  the  density  of  the  air,  and  considering  the 
strata  as  horizontal,  for  the  reason  that  it  is  seldom  practicable 
to  determine  the  effect  of  such  irregularities  with  precision. 

The  general  theory  of  astronomical  refraction,  as  it  will  be  set 
forth  in  the  present  chapter,  therefore  rests  on  the  assumption 
that  in  the  air  the  strata  of  equal  density,  or  the  equiponderant 
strata,  are  horizontal ;  and  that  the  density  continually  di- 
minishes from  the  earth  to  the  outer  limit  of  the  atmosphere, 


174  ASTRONOMICAL  EEFRACTION  [§  92. 

subject  to  the  conditions  of  equilibrium.  What  that  limit  may 
be  is  not  yet  known ;  we  can,  however,  say  with  confidence  that 
above  a  height  of  60  kilometres  the  atmosphere  is  so  rare  that 
its  refractive  power  need  not  be  investigated. 

Refraction  may  be  treated  in  three  sections.  In  the  present 
section  the  law  of  density  of  the  air  on  which  its  refractive 
power  depends  will  be  developed.  In  the  next  section  a  general 
conspectus  of  the  laws  and  results  of  refraction  by  the  atmos- 
phere will  be  put  into  an  elementary  form.  The  third  section 
will  treat  the  general  theory  of  astronomical  refraction  proper. 

93.  Density  of  the  atmosphere  as  a  function  of  the  height. 

We  put  in  this  section 

p,  the  density  of  the  atmosphere  at  any  height. 

pv  the  density  at  the  earth's  surface,  or  at  the  point  where 
the  observer  is  situated. 

p0,  the  "standard  density"  under  standard  barometric  pressure 
(760  mm.  at  Paris)  and  temperature  0°  C. 

g,  the  ratio  of  the  intensity  of  gravity  at  any  point  to- 
standard  gravity. 

p,  the  pressure  of  the  air  per  unit  of  surface  at  any  altitude. 

h,  altitude  above  the  surface  in  linear  measure. 

r,  the  temperature  in  degrees  centigrade  above  absolute  zero. 

TO,  the  absolute  temperature  of  the  centigrade  0°. 

It  should  be  remarked  that  the  temperature  from  which  r  ia 
counted  is  not  strictly  the  absolute  zero,  but  the  temperature  at 
which  the  volume  of  air  would  become  zero,  supposing  it,  when 
placed  under  constant  pressure,  to  continue  its  diminution  of 
volume  with  falling  temperature  at  the  same  rate  that  it  is 
observed  to  vary  through  the  range  of  temperature  at  which 
refraction  occurs.  This  variation  of  volume  with  temperature  i& 
assumed  to  be  linear,  in  accordance  with  the  law  of  Gay  Lussac. 

If  we  measure  the  coefficient  of  expansion  by  the  expansion 
at  0°  C.  produced  by  a  change  of  1°  C.  in  the  temperature,  the 
absolute  zero  as  here  used  will  be  the  negative  reciprocal  of  the 
coefficient.  The  following  four  values  of  the  coefficient  and  of 
the  absolute  zero  are  the  results  of  numbers  determined  or 


§94.]  DENSITY  OF  THE  ATMOSPHEEE  175 

adopted  by  different  authorities.  The  first  two,  those  of  Regnault 
and  Mendelejeff,  are  determined  by  experiments  in  the  laboratory. 
The  third  is  that  adopted  by  Bessel  in  his  tables  of  refraction. 
The  fourth  is  that  adopted  in  the  Poulkova  tables  of  refraction. 

Coef.  of  exp.  Abs.  zero. 

Kegnault  -003670  -272°-5 

Mendelejeff  -003684  -271-4 

Bessel  (adopted)  -003  643  3  -  274  -5 

Poulkova  (adopted)  -003689  -271  -1. 

The  mean  of  the  experimental  determinations  would  be 
—  271°'9.  But,  as  the  actual  amount  of  the  refraction  may  be 
affected  by  the  aqueous  vapour  in  the  atmosphere,  we  shall  give 
equal  weight  to  the  number  adopted  at  Poulkova  and  to  that 
derived  by  experiment,  thus  taking  for  the  temperature  in 
question 

Temp.  C.  of  abs.  zero=  -271C<5. 

The  expression  for  T  in  terms  of  temperature  C.  is  therefore 
r  =  temp. 


94.  The  following  two  laws  of  physics  are  accepted  as  the 
basis  of  the  subject. 

First  law  :  The  pressure  per  unit  of  surface  due  to  the 
elasticity  of  the  air  is  proportional  to  the  product  of  the  density 
of  the  air  into  its  absolute  temperature. 

Second  law:  Along  any  vertical  line  in  the  atmosphere  the 
diminution  of  the  pressure  through  any  height  is  equal  to  the 
weight  of  a  stratum  of  air  of  unit  base  and  of  that  height. 

The  first  of  these  laws  cannot  be  true  for  temperatures 
approaching  the  absolute  zero,  and  the  density  also  varies 
slightly  from  the  law  for  very  high  temperatures.  But  these 
deviations  are  unimportant  in  the  theory  of  refraction.  At  an 
altitude  where  there  is  any  possibility  of  the  temperature 
approaching  absolute  zero  the  air  is  so  rare  as  to  be  without 
appreciable  effect  upon  the  refraction.  On  the  other  hand, 
astronomical  observations  are  not  made  at  any  but  ordinary 
temperatures,  so  that  the  deviation  in  the  case  of  high  tempera- 
tures need  not  be  considered. 


176  ASTRONOMICAL  REFRACTION  [§94. 

The  second  law  when  rigorously  applied  requires  us  to  take 
account  of  the  diminution  of  gravity  with  height,  which  makes 
the  actual  law  of  density  somewhat  different  from  what  it 
would  be  were  gravity  equal  at  all  heights.  But,  as  we  shall 
hereafter  see,  this  deviation  is  without  important  influence  upon 
the  refraction. 

The  first  law  may  be  expressed  in  the  form  of  the  equation 

P  =  yrp,  .................................  (1) 

y  being  a  constant  depending  on  the  elasticity  of  air  at  a  given 
temperature  and  density.  The  density  p  being  the  fundamental 
quantity  on  which  the  refraction  depends,  this  law  may  be 
expressed  for  practical  use  in  the  form 


In  developing  this  subject  it  is  important  to  define  the  units 
in  which  the  various  quantities  are  to  be  expressed.  Since  the 
-element  of  time  does  not  enter  into  the  theory,  and  the  element 
of  mass  enters  only  in  a  subsidiary  role,  the  most  convenient 
units  will  not  be  those  of  the  C.G.S.  system,  but  the  following  : 

The  unit  of  length  :  arbitrary. 

The  unit  of  volume  :  the  cube  whose  side  is  the  unit  of  length. 

The  unit  of  weight  :  the  weight  of  unit  volume  of  standard 
water  under  standard  gravity.  For  standard  gravity  it  will  be 
convenient  to  take  gravity  at  Paris. 

The  unit  of  pressure  :  the  pressure  of  unit  weight  upon  unit 
surface. 

The  result  of  this  choice  of  units  is  that  the  constant  y  and 
the  pressure  h  may  both  be  defined  as  lengths.  It  will  be  seen 
that  the  constant  y  may  be  defined  as  the  elastic  pressure  of 
the  air  at  the  absolute  temperature  1°  when  compressed  to  the 
density  of  water.  Assuming  that  the  first  law  can  be  continued 
to  this  temperature,  the  constant  may  be  yet  better  apprehended 
by  considering  it  as  a  height  of  the  column  of  standard  water 
which,  under  gravity  at  Paris,  would  condense  air  at  absolute 
temperature  1°  to  density  1. 

It  is  also  convenient  to  adopt  a  standard  of  temperature,  to 


§94.]  DENSITY   OF  THE  ATMOSPHERE  177 

which  all  the  fundamental  data  will  be  referred.  The  best 
standard  for  tables  of  refraction  is  one  not  differing  greatly  from 
the  mean  temperature  at  which  astronomical  observations  are 
made.  A  convenient  standard  is  10°  C.  or  50°  F.  But,  in  in- 
vestigating the  theory  of  the  subject,  it  is  better  to  take  0°  C., 
because  this  is  the  standard  temperature  for  nearly  all  physical 
investigations  and  numerical  constants  which  enter  into  the 
theory.  When,  therefore,  we  speak  of  760  mm.  of  mercury  as 
a  standard  pressure,  we  mean  mercury  at  0°  C.,  to  which  we 
suppose  the  observed  height  to  be  reduced  by  correcting  it  for 
the  temperature  of  the  mercury. 

As  our  units  have  just  been  defined,  the  weight  w  of  any 
volume  v  of  air  will  be  given  by  the  equation 


Now,  consider  a  prism  of  air  of  unit  surface  and  of  infinitesimal 
height  dh.  The  weight  of  this  prism  is  gpdh.  Hence,  regarding 
the  prism  as  horizontal,  the  change  of  pressure  through  an 
infinitesimal  change  in  height  is  given  by  the  equation 

dp  =  -gpdh,  ..............................  (3) 

the  sign  being  negative  because  the  pressure  decreases  as  we 
ascend. 

If  we  regard  p  as  constant  through  the  height  h  of  a  column 
of  air,  the  pressure  at  the  base  of  this  column  produced  by  its 
weight  will  be 

p  =  9ph  ..................................  (4) 

A  comparison  of  this  equation  with  (1)  will  give  us  the  height 
of  a  column  of  air  which,  under  standard  gravity,  would  produce 
the  same  pressure  that  is  actually  exerted  by  the  entire  body  of 
the  air  at  the  base  of  the  column.  This  height  is  called  the 
pressure-height.  Let  us  put 

hv  the  pressure-height. 

Substituting  this  value  of  h  in  (4)  and  comparing  with  (1),  we 
see  that  the  pressure  height  is  given  by  the  equation 


N.S.A. 


178  ASTKONOMICAL  KEFR  ACTION  [§  94. 

The  three  quantities  p,  p,  and  T  all  vary  with  the  height  h 
above  the  surface,  and  are  to  be  expressed  as  functions  of  h. 
Regarding  them  as  such,  the  differentiation  of  (2)  gives 

dp_    (dp       dr\  a. 

-- 


By  substituting  the  value  of  dp  from  (3),  we  now  have,  after 
simple  reductions, 


dh 


which   is   the  fundamental  equation  for  the  diminution  of  the 
density  of  the  air  with  its  height. 

95.  Numerical  data  and  results. 

It  will  conduce  to  clearness  if,  before  going  farther,  we 
consider  the  numerical  values  of  the  fundamental  quantities. 

According  to  Regnault,  whose  results  we  accept  for  this 
purpose,  the  density  of  air  (water  =  1)  at  0°  C.  (T  =  27l°-5  =  T0) 
and  under  a  barometric  pressure  of  760  mm.  (gravity  at  Paris)  is 

Po  =  0-001  293  2. 

This  value  is  substantially  confirmed  by  the  more  recent  ones 
of  Rayleigh  and  of  Leduc. 

Taking  the  metre  as  the  unit  of  length,  the  unit  of  pressure, 
as  we  have  defined  it,  will  be  the  weight  at  Paris  of  a  cubic 
metre  of  water  pressing  upon  a  square  metre  of  surface.  The 
standard  barometric  pressure  just  cited  will  be 

Density  of  quicksilver  x  O760. 

Taking  for  this  density  13*596,  this  product  is  10-333,  the 
standard  pressure.  Substituting  this  value  of  p,  and  the  values 
of  T0  and  pQ  corresponding  to  the  given  conditions  in  the  equation 
(1),  we  have 

10-333  =  y  x  271-5  x  O'OOl  293  2, 

which  gives  y  =  29'429  metres (8) 

This  value  of  y  being  substituted  in  (2)  and  (7)  will  give  p  and 
in  terms  of  p,  g,  and  T.     It  will,  however,  be  convenient  to 


§  97.]  NUMERICAL  DATA  AND  RESULTS  179 

substitute   for  p  the   height  of  the   barometer,  by   which  p  is 
determined  in  practice.     Let  us  then  put 

6,  the   barometric  reading  -r-  760  mm.  and  corrected  for   tem- 
perature. 

We  shall  then  have         p  =  10-33360, 
and  the  equations  (2)  and  (7)  may  be  written 

P  =  -0012932^%,  ...........................  (9) 

0-351  10  , 


dh 


.(10) 


96.  An  interesting  conclusion  may  be  drawn  from  this  last 
equation.  If  the  rate  of  diminution  of  temperature  with  height 
is  such  that 


r=-  0-033 

which,  g  never  differing  much  from  1,  implies  a  diminution  at 
the  rate  of  1°  in  about  30  metres,  p  will  remain  constant,  the 
effect  of  continually  diminishing  pressure  being  compensated  by 
the  diminishing  temperature.  If  this  constant  rate  of  diminution 
continues  we  shall,  at  the  pressure-height,  have  r  =  0,  at  which 
point  the  atmosphere  will  cease.  That  is  to  say,  the  result  will 
be  that  the  atmosphere  would  become  an  ocean  of  uniform 
density  with  a  definite  upper  surface. 

97.  General  view  of  requirements. 

The  logical  order  of  the  requirements  for  our  theory  is  this  : 
In  order  to  determine  the  refraction  at  considerable  zenith 
distances  it  is  necessary  to  have  an  expression  for  the  density  of 
the  air  as  a  function  of  the  height.  This  density  does  not  admit 
of  direct  observation  ;  it  must  therefore  be  inferred  from  the  law 
of  diminution  of  temperature  with  height.  When  this  is  known, 
the  law  of  density  is  derived  by  substituting  in  the  equation  (7) 

the  values  of  r  and  of  -^,  which  will  be  functions  of  h.     The 

dh 


180  ASTRONOMICAL  REFEACTION  [§  97. 

integration  of  the  equation  will  then  give  the  expression  for  the 
density  at  any  height. 

The  diminution  of  temperature  with  height  is  of  such  a 
character  that  it  cannot  be  exactly  represented  by  any  formula. 
It  varies  with  the  time  of  day,  the  seasons,  the  temperature  at 
the  surface  of  the  earth,  and  the  height  itself.  All  we  can  do, 
therefore,  is  to  construct  some  hypothetical  formula  which  will 
give  a  result  as  near  as  possible  to  the  general  average.  This 
formula  may  be  based  partly  on  theoretical  considerations 
showing  what  laws  of  diminution  are  more  or  less  probable  and 
on  observations  with  the  aid  of  kites  and  balloons.  These 
observations  have  been  greatly  extended  during  the  past  few 
years,  and  the  results  enable  us  to  formulate  laws  of  temperature 
with  much  greater  confidence  than  was  formerly  possible. 

Some  theoretical  considerations  will  help  to  guide  us.  Were 
the  atmosphere  in  a  state  of  complete  rest  the  general  theory  of 
heat  leads  to  the  conclusion  that  it  would  tend  to  assume  the 
same  temperature  throughout.  This  tendency  may  be  counter- 
acted in  one  direction  or  another  by  the  effect  of  possible  thermal 
coloration  of  the  air,  a  subject  about  which  not  enough  is  known 
to  form  the  basis  of  a  conclusion.  A  state  of  constant  tempera- 
ture has  also  been  shown  by  Tait  to  result  from  the  kinetic 
theory  of  gases.  This  state  is  therefore  called  one  of  thermal 
equilibrium. 

But  the  atmosphere  is  not  at  rest,  being  subject  to  ascending 
and  descending  currents.  Whenever  a  body  of  air  ascends,  it 
expands  and  thereby  cools.  When  a  body  descends  it  is  com- 
pressed into  smaller  space  and  thereby  becomes  warmer.  If  the 
air  were  constantly  stirred  from  top  to  bottom,  a  condition 
would  be  reached  in  which,  when  any  body  of  it  ascended,  it 
would,  as  it  expanded  and  cooled,  constantly  be  at  the  same 
temperature  as  the  surrounding  air.  This  condition  is  that  of 
adiabatic  equilibrium. 

When  an  atmosphere  in  thermal  equilibrium  is  stirred  so  as 
to  bring  it  nearer  the  state  of  adiabatic  equilibrium,  work  must 
be  done.  For  whenever,  in  such  a  case,  a  body  of  air  is  lifted 
up,  it  will  by  its  expansion  be  colder,  and  therefore  denser,  than 


§98.]  GENERAL  VIEW  OF  REQUIREMENTS  181 

the  surrounding  air.  It  will,  therefore,  tend  to  fall  again,  and 
work  must  be  done  in  order  to  raise  it.  A  body  of  air  falling 
becomes  warmer,  and  so  rarer,  than  the  surrounding  air,  thus 
tending  to  rise  again.  We  conclude  that  the  centre  of  gravity 
of  a  column  of  air  is  higher  in  the  case  of  adiabatic  than  in 
that  of  thermal  equilibrium. 

The  law  of  diminution  of  temperature  in  the  case  of  adiabatic 
equilibrium  is  such  that  the  diminution  of  temperature  with 
height  is  at  the  rate  of  about  10°  C.  per  kilometre. 

If  the  only  source  from  which  the  air  obtained  heat,  or  to 
which  it  communicated  heat,  were  the  ground  on  which  it  rests, 
the  tendency  of  ascending  and  descending  currents  would  be 
toward  bringing  about  a  condition  of  adiabatic  equilibrium  of 
temperature  throughout.  But  this  tendency  is  continually 
counteracted  by  the  effect  of  the  sun's  rays,  and  by  radiation 
from  the  warmer  portions  of  the  air  to  the  cooler  portions.  It 
is  a  result  of  the  laws  of  radiation  in  space  that,  at  each  distance 
from  the  sun,  there  is  a  certain  definite  temperature  which  a 
neutral  coloured  body,  or  a  body  for  which  the  reflecting  and 
absorbing  powers  were  the  same  for  all  wave-lengths,  would 
reach.  This  normal  temperature  could  be  fairly  well  determined, 
and  has  especially  been  investigated  by  Poynting,  his  conclusions 
being  based  on  the  law  that  the  emission  of  heat  by  a  warm 
body  is  proportional  to  the  fourth  power  of  its  absolute  tem- 
perature. 

It  follows  that,  except  so  far  as  influenced  by  thermal 
coloration,  portions  of  the  atmosphere  which  are  below  this 
normal  temperature  will  be  warmed  toward  it  by  absorption  of 
the  sun's  rays,  while  those  which  are  at  a  temperature  above 
the  normal  will  lose  heat  by  their  own  radiation.  Thus  arises  a 
tendency  toward  thermal  equilibrium  the  exact  extent  of  which 
cannot  be  determined  by  theory,  but  only  by  observation. 

98.  Density  at  great  heights. 

The  probable  upper  limit  of  the  atmosphere  may  be  considered 
in  this  connection.  Observations  of  meteors  and  shooting  stars 
seem  to  show  that  these  bodies  are  seen  at  an  altitude  of  more 


182  ASTRONOMICAL  REFRACTION  [§  98. 

than  100,  possibly  200,  miles.  The  conclusions  from  this  would 
be  that  the  rate  of  diminution  of  temperature  must  diminish, 
so  that  the  absolute  zero  is  never  reached. 

Another  indication  of  the  height  of  the  atmosphere  is  afforded 
by  twilight.  This  terminates  when  the  sun  is  at  a  depression 
of  between  15°  and  18°  below  the  horizon.  This  leads  to  the 
conclusion  that  the  power  of  reflecting  light  ceases,  owing  to 
the  rarity  of  the  atmosphere,  at  a  height  of  about  70  kilometres. 
This  fact  suggests  that  the  variations  of  temperature  of  these 
high  regions  of  the  air  at  different  latitudes  and  seasons  is 
probably  less  than  near  the  surface  of  the  earth.  Altogether 
theory  can  tell  us  little  about  the  actual  diminution  of  the 
temperature  with  height  except  that  it  is  materially  less  than 
10°  C.  per  kilometre.  We  must,  therefore,  derive  an  empirical 
law  of  diminution  from  observations. 

The  mass  of  material  here  at  our  disposal  is  very  great,  and, 
belonging  to  the  province  of  meteorology,  cannot  be  considered 
in  the  present  wrork.  The  most  valuable  of  this  material 
consists  of  observations  made  by  kites  or  balloons  carrying  self- 
registering  thermometers  to  the  greatest  possible  height.  For 
several  years  past  extensive  kite  observations  have  been  made 
at  the  Blue  Hill  Observatory,  Hyde  Park,  Mass.,  by  Dr.  A.  L. 
Rotch.  An  extended  system  both  of  kite  and  balloon  observa- 
tions is  being  carried  out  by  the  U.S.  Weather  Bureau.  In 
Germany  and  France  balloons  have  recently  been  successfully 
sent  to  a  height  of  12  or  more  kilometres.  Without  going  into 
unnecessary  details,  it  may  be  said  that  these  observations, 
notwithstanding  the  irregularities  naturally  inherent  in  the 
subject,  lead  to  the  following  general  conclusions  : 

1.  The  annual  and  diurnal  changes  of  temperature  diminish 
with  the  height,  both  becoming  very  small  at  the  highest  attain- 
able altitudes.      This  result  points  to  the  conclusion  that  the 
sun's  rays  have  but  little  immediate  influence  on  the  temperature 
of  the  higher  strata  of  the  atmosphere.     Another  obvious  result 
is  that  the  fall  of  temperature  must  be  more  rapid  the  higher  the 
temperature  is  at  the  surface  of  the  earth. 

2.  The  diminution  of  the  diurnal  variation  of  temperature  is 


§  99.]  DENSITY  AT  GREAT  HEIGHTS  183 

very  rapid  near  the  earth's  surface,  being  reduced  to  one  half 
.at  an  altitude  of  a  few  hundred  metres.  One  result  of  this 
is  that  during  the  day  the  diminution  is  very  rapid  at  low 
altitudes,  but,  during  the  night,  especially  the  later  hours  of  the 
night,  is  changed  to  an  actual  increase. 

3.  The  general  average  diminution  near  the  ground,  day  and 
night,  taking  the  whole  year  round,  is  not  far  from  6°'5  C.  per 
kilometre.  Astronomical  observations  being  mostly  made  at 
night,  the  diminution  for  them  will  be  less.  Were  the  rate  of 
•diminution  near  the  surface  of  the  earth  important,  it  would  be 
necessary  to  suppose  a  very  small  rate  in  the  lowest  kilometre 
<of  the  air  for  the  purpose  of  computing  the  astronomical  refrac- 
tion for  night  observations.  But,  for  reasons  which  will  be 
better  understood  when  the  general  theory  is  developed,  astro- 
nomical refraction  is  little  influenced  by  the  diminution  of 
temperature  at  low  altitudes,  the  effect  of  differences  of  tempera- 
ture reaching  their  maximum  near  the  pressure- height,  and 
.slowly  diminishing  for  yet  greater  heights.  We  must,  therefore, 
for  astronomical  purposes,  lay  more  stress  on  the  temperature 
.at  considerable  heights  than  near  the  surface  of  the  earth. 
This  will  enable  us  to  include  an  expression  for  all  heights  in 
some  simple  formula. 

99.  Hypothetical  laws  of  atmospheric  density. 

The  various  tables  of  refraction  which  have  been  constructed 
for  astronomical  use  rest  upon  different  expressions  for  the 
density  of  the  air  as  a  function  of  the  altitude.  It  will  also  be 
instructive  to  consider  hypothetical  laws  of  diminution  which 
.are  the  simplest  in  form.  Amongst  the  various  hypotheses  that 
have  been  made,  or  may  be  made,  the  following  are  worthy 
of  citation : 

A,  the  hypothesis  of  constant   temperature  at  all  altitudes. 
This  hypothesis  was  adopted  by  Newton,  and  is,  therefore,  asso- 
ciated with  his  name.     The  law  to  which  it  leads  is  extremely 
simple. 

B,  the  hypothesis  that  the  density  diminishes  uniformly  with 
the  height.     This  is  one  of  the  forms  which  a  law  proposed  by 


184  ASTRONOMICAL  REFRACTION  [§99. 

Bouguer,  and  adopted  by  Simpson  and  Bradley,  may  take.  On 
this  hypothesis  the  density  becomes  zero,  and  the  atmosphere 
reaches  its  limit  at  twice  the  pressure-height,  or  an  altitude  of 
about  16  kilometres.  This  is  obviously  too  low ;  yet  the  hypo- 
thesis gives  a  better  approximation  to  the  truth  so  far  as 
refraction  is  concerned,  than  that  of  constant  temperature. 

C,  Bessel's  Hypothesis.     This  is  a  modified  form  of  the  hypo- 
thesis  of   Newton.     It  is   not  based    on    any  assumed   law    of 
temperature,  but  is  an  expression  for  the  density  in  terms  of  the 
altitude.     As   will  be  shown  hereafter,  it  is  not  altogether  ad- 
missible. 

D,  the  hypothesis  that  the  temperature  diminishes  at  a  uniform 
rate  with  the  height  at  all  heights.     This  is  in  accordance  with 
the  law  of  adiabatic  equilibrium,  and  accords  most  nearly  with 
the  results  of  the  highest  balloon  observations,  which  show  no 
falling  off  in  the  rate  of  diminution  at  the  great  heights  yet 
attained.     But  it  is  not  accordant  with  the  kite  observations, 
which,  in  the  general   average,   seem  to  show  a  falling  off  of 
the   rate   at   an   altitude   of   a   very   few   kilometres.     It   was 
developed  by  Ivory,  with  whose  name  it  may  be  associated. 

On  this  hypothesis,  using  the  most  likely  rate  of  diminution, 
the  absolute  zero  would  be  reached,  and  the  atmosphere  would 
have  a  limit  at  a  height  of  50  kilometres  more  or  less. 

E,  the  hypothesis  that  the  temperature  diminishes  by  a  con-' 
stant  fraction  of  its  absolute  amount  for  every  unit  of  increase 
in  the  altitude.     The  three  conditions  which  this  law  satisfies  are 
that  of  a  rate  of  diminution  which  shall  be  more  rapid  in  warm 
weather  than  in  cold,  and  which  shall   slowly  diminish    with 
increasing  height.     It  fulfils  the  additional  condition  that  the 
absolute  zero  shall  not  be  reached  at  any  finite  altitude.     Yet, 
it  cannot  be  applied  so  as  to  be  strictly  consonant  with   the 
temperature  resulting  from  balloon  ascensions. 

It  may  be  remarked  that  the  preceding  hypotheses  are  not 
entirely  distinct,  A  and  B  being  really  special  cases  of  Ivory's 
hypothesis.  When,  in  D,  the  diminution  of  temperature  is 
reduced  to  zero,  we  have  on  this  hypothesis  a  constant  tempera- 
ture, and  therefore  Newton's  hypothesis.  If  the  temperature 


§  100.]  LAWS   OF   ATMOSPHERIC   DENSITY  185 

diminishes  at  such  a  uniform  rate  as  to  reach  the  absolute  zero 
at  the  pressure  height,  we  shall  have  the  law  B.  The  truth 
probably  lies  between  these  two  very  wide  extremes. 

100.  Development  of  the  hypotheses. 

So  far  as  refraction  is  concerned,  we  can  assign  a  rate  of 
diminution  on  either  of  the  last  two  hypotheses  which  shall  bring 
their  results  into  close  agreement  at  moderate  heights.  When 
this  is  done  it  is  probable  that  either  of  them  will  represent  the 
law  of  refraction  equally  well.  Our  choice  between  them  or  our 
combination  of  them  must,  therefore,  be  a  matter  of  convenience. 
We  shall  now  show  the  formulae  for  density  to  which  these 
various  hypotheses  lead  : 

A.  Newton  's  hypothesis  of  constant  temperature.  Disregarding 
the  diminution  of  gravity  with  altitude,  and  supposing  the  tem- 
perature constant  (7)  gives 

dp=-y=      1 
pdh      -yr          hi 

By  integration,         log  p  =  C~j-, 

fli 

C  being  the  arbitrary  constant  of  integration.     Putting 

plt  the  density  at  the  surface  of  the  earth,  or  at  the  point  of 
observation, 
we  shall  have  C= 


_ 
and  p  =  ple~hi  ...............................  (11) 

The  best  idea  of  this  result  will  be  gained  by  the  consideration 
that  it  shows  the  density  to  diminish  in  geometrical  progression 
as  the  altitude  increases  in  arithmetical  progression.  A  clear 
general  conception  of  the  law  may  be  gained  by  finding  an 
altitude  h0,  at  which  the  density  is  reduced  to  one  half.  We  may 
take  8  kilometres  as  the  usual  value  of  hv  the  pressure-height. 
We  then  find  A0  by  putting,  in  (11),  p  =  -|  ;  pl  ;  /^  =  8. 

*0 

68=2, 

which  gives  h0  =  8  Nap.  log  2  =  5'54, 

h0  being  expressed  in  kilometres.     The  density  would,  therefore, 

be  reduced  one-half  for  about  every  5J  kilometres,  or  3J  miles. 


186  ASTRONOMICAL  REFRACTION  [§  100. 

Since  210=1024   the   density   is  reduced  to  "001  of  its  amount 
at  the  height  of  56  kilometres,  or  35  miles. 

Hypothesis  B.     On   this   hypothesis   the   expression   for   the 
density  assumes  the  very  simple  form 


(12) 


Although,  as  already  pointed  out,  this  hypothesis  fixes  the 
limit  of  the  atmosphere  at  too  low  a  point,  it  is,  so  far  as  the 
effect  on  refraction  is  concerned,  markedly  nearer  the  truth  than 
that  of  constant  temperature.  The  actual  truth  lies  between  the 
two  but  nearer  to  B  than  to  A. 

C.  BesseUs  Hypothesis.  Bessel,  disregarding  any  law  based 
on  temperature,  proposed  a  law  of  density  of  the  same  general 
form  with  the  exponential  one  of  constant  temperature,  the 
exponent  being  multiplied  by  a  factor  less  than  unity.  Although, 
in  the  most  general  form  of  its  statement,  it  was  implied  that 
the  factor  might  vary  with  the  height,  in  practical  use  the 
constant  factor 

k  =  0-9649, 

was  used.      With   this   factor   the   expression   for   the   density 
assumes  the  form 

P  ~^iT  /T  O\ 

—  =  e    fci (13) 

On  this  law  was  based  the  tables  of  refraction  published  in  the 
Tabulae  Regiomontanae,  which  have  been  in  extensive  use  even 
up  to  the  present  time. 

There  is  no  law  of  variation  of  temperature  which  will 
correspond  to  this  law  of  density.  The  fact  is  that,  as  a  law,  it 
is  not  possible.  Every  possible  law  of  variation  must  make  the 
pressure  of  the  entire  atmosphere  equal  to  the  pressure  at  the 
base.  In  order  that  this  result  may  follow,  it  is  necessary  that 
the  definite  integral  which  expresses  the  entire  weight  of  the 
atmosphere  shall  give  the  basic  pressure.  This  requirement  is 
expressed  by  the  condition 

f30 

pdh  =  pressure  at  base  =  hlpr 


§  100.]  LAWS  OF  ATMOSPHERIC   DENSITY  187 

But,  by  integrating  the  equation  (13),  it  is  seen  that  the 
pressure  at  the  base  of  the  atmosphere  becomes,  on  Bessel's 
hypothesis. 


which  is,  therefore,  too  great  by  between  three  and  four  per  cent. 
of  its  entire  amount. 

D.  Ivory's  hypothesis  of  uniform  diminution  of  temperature 
with  altitude.  Taking  the  hypothesis  of  diminution  of  tem- 
perature at  a  uniform  rate  with  the  altitude,  and  assuming  the 
rate  to  be  proportional  to  the  temperature  at  the  base,  the 
expression  of  the  temperature  in  terms  of  the  altitude  and  of 
the  temperature  TJ  at  the  base  will  be  of  the  form 


j8  being  a  constant  factor.     The  constant  rate  of  diminution  is 
then 


and  the  equation  becomes 


pdh 
The  integration  of  this  equation  gives 

log  p  =  log  Pi+9~p^1  log(l  -  , 
log  /o1  being  the  arbitrary  constant,  so  taken  that,  for  h  =  0,  p  =  pr 

If  we  put  y 

this  equation  will  give 


PI 

One  result  of  this  law  is  that  the  atmosphere  would  terminate 
at  the  altitude  for  which  /3h  =  l,  If  we  put  A0  for  this  altitude, 
the  preceding  expression  may  be  written 


188  ASTRONOMICAL  REFRACTION  [§  100. 

The  general  average  result  of  balloon  observations  is  a  rate  of 
diminution  which  would  place  the  absolute  zero  at  the  height  of 
about  50  km.  The  exponent  y  may  be  written,  putting  </  =  !, 


This  would  give  y  =  5  for  a  surface  temperature  of  12°  C.,  but, 
the  exponent  being  a  function  of  the  temperature,  will  in  general 
be  a  fractional  quantity.  For  0°  C.,  we  have,  very  nearly, 

y  =  5-25. 

Hyp.  E.  Diminution  of  temperature  in  constant  geometrical 
progression  with  altitude.  If  we  put 

/3,  the  rate  of  proportional  decrease. 

The  expression  for  the  rate  of  diminution  of  the  temperature 
in  terms  of  the  altitude  will  be 

^    £--*•  ...............................  <18> 

r  being  the  temperature  at  any  point. 
Integration  gives  log  —  =  fih, 

or  -  =  e&t   .............................  (19) 

Ti 

as  a  relation  between  the  temperature  of  any  two  points  on  the 
same  vertical  line  at  altitudes  differing  by  h. 

Accepting  this  law  of  temperature,  the  equation  (7)  shows  the 
corresponding  law  for  the  density.  Substituting  (18)  in  (7),  we 

have  -&  =  B  __  *Lef* 

p  dk  yTl 

T!  being  the  temperature  for  k  =  0. 
Integration  gives  ph+c_    ff.  __ 


C  being  the  arbitrary  constant  of  integration.     Putting  pv  the 
temperature  for  h  =  0,  we  have 


Thus  the  expression  for  the  density  becomes 

^-^-(^-l)  ...................  (20) 

Pi  PVTi 


.§  101.]  LAWS  OF  ATMOSPHERIC    DENSITY 

Representing  this  quantity  by  —v,  we  have 


Pi 


where 


189 

.(21) 
.(22) 


This  expression  is  too  complex  for  convenient  use,  unless  the 
process  of  integrating  the  differential  of  the  refraction  is  made 
largely  numerical. 

As  it  may  be  of  interest  to  compare  the  density  at  different 
heights  on  the  two  hypotheses,  the  following  table  has  been 
prepared.  In  each  case  there  are  two  constants  to  which  values 
at  pleasure  may  be  assigned.  One  is  the  temperature  ^  at  the 
station ;  the  other  the  rate  of  diminution  with  altitude. 

101.  Comparison  of  densities  of  the  air  at  different  heights  on 
hypotheses  D  and  E. 

In  D  as  used,  the  limit  of  the  atmosphere  is  taken  to  be 
48  km.,  and  the  rate  of  diminution  of  temperature  to  be  T^-r-48, 
which,  for  0°  C.,  is  nearly  5°'5  per  km.  On  hypothesis  E  the 
rate  of  diminution  is  taken  to  be  rH-45  throughout.  This  is 
6°  C,  very  nearly,  at  0°  C. 


HYP.  D. 

HYP.  E. 

E-D. 

^  =  261°  -8 

^  =  272°  -6 

7-!  =  284°  -6 

^=246°  -5 

r1-271°-5 

T-^2960  -5 

271°  -5 

h 

«!=  -9°  -7 

«1=  +  r-i 

<1=  +  13°-1 

<!  =  -  25° 

^  =  0° 

«!  =  +  25° 

^  =  0° 

0 

1-00000 

1-00000 

1-00000 

1-00000 

1-00000 

1  -00000 

0 

5 

0-56130 

0-57694 

0-59310 

0-53426 

0-57373 

0-60347 

-  -00167 

10 

0-29335 

0-31095 

0-32420 

0-25489 

0-30343 

0-33815 

-  -00603 

15 

0-13982 

0-15368 

0-16870 

0-11658 

0-14026 

0-17528 

-  -01203 

20 

0-05904 

0-06754 

0-07729 

0-04635 

0-06508 

0-08280 

-  -00158 

25 

0-02102 

0-02526 

0-03021 

0-01637 

0-02657 

0-03538 

+  -00175 

30 

0-00580 

0-00742 

0-00948 

0-00504 

0-00896 

0-01347 

+  -00172 

35 

0-00105 

0-00146 

0-00200 

0-00134 

0-00273 

0-00452 

+  -00132 

40 

0-00008 

0-00013 

0-00020 

0-00030 

0-00071 

0-00129 

+  -00059 

190  ASTRONOMICAL  REFRACTION  [§  102. 

Section  II.    Elementary  Exposition  of  Atmospheric  Refraction. 

102.  It  is  a  general  law  of  optics  that  the  course  followed  by 
a  ray  of  light  is  along   the  same  line  or  curve   for  the  two 
opposite  directions  in  which  the  light  may  move.     It  follows 
that  the  course  of  a  ray  of  light  from  a  celestial   object  may 
equally  well  be  studied  as  that  of  a  ray  passing  from  the  observer 
to  the  object.     Since,  in  practical  astronomy,  the  given  quantity 
is  commonly  the  direction  of  the  ray  when  it  reaches  the  ob- 
server, this  reverse  direction  is  generally  the  simplest  to  consider. 
But  the  differential  equations  are  the  same  in  both  cases. 

The  apparent  zenith  distance  of  a  celestial  object  is  that  of 
the  ray  when  it  reaches  the  observer.  The  true  zenith  distance 
is  the  angle  which  the  ray  makes  with  the  vertical  of  the 
observer's  station  before  it  enters  the  atmosphere.  We  readily 
see  that  the  curvature  of  the  ray  is  always  concave  toward  the 
earth,  so  that  the  effect  of  refraction  is  always  to  make  the 
apparent  less  than  the  true  zenith  distance.  Hence  the  cor- 
rection to  the  Z.D.  for  refraction  is  always  positive. 

The  atmospheric  strata,  being  always  perpendicular  to  the 
direction  of  the  vertical  at  the  point,  are  separated  by  curved 
surfaces,  of  which  the  curvature  is  determined  by  that  of  the 
geoid.  Practically  they  may  be  treated  as  spherical  above  any 
one  region  of  the  earth.  If  the  zenith  distance  is  small,  the 
curvature  is  so  slight  that  the  refraction  will  then  be  nearly  the 
same  as  if  the  surfaces  of  the  strata  were  planes.  A  close 
approximation  to  the  refraction  at  moderate  zenith  distances 
may,  therefore,  be  obtained  on  this  hypothesis. 

103.  Refraction  at  small  zenith  distances. 

The  following  theorem  forms  the  starting  point  in  the 
subject : 

Regarding  the  atmospheric  strata  as  plane  and  parallel,  the 
total  amount  of  refraction  is  independent  of  the  law  of  diminu- 
tion of  the  refractive  power  with  height,  and  depends  solely  upon 
the  refractive  power  of  the  air  at  the  surface  of  the  earth. 


§  103.]      REFRACTION   AT   SMALL   ZENITH   DISTANCES  191 

To  show  this  we  put  : 

/x0,  /ULI}  fjL2,  .  .  .  /u.n,  the  indices  of  refraction  of  any  number  of  suc- 
cessive atmospheric  strata,  from  the  outer  limit  of  the  atmos- 
phere to  the  earth,  which  values  start  from  yu0  =  l  as  their 
beginning. 

z0)  zlt  zz,  ...  zn,  the  angles  of  incidence  at  which  the  ray  enters 
the  successive  strata,  which  are  the  same  as  those  of  refraction 
at  which  it  leaves  the  bottom  of  each  stratum  above. 

Then,  by  the  law  of  refraction,  we  have 


sn  z  :  sn  z      = 


sin  sn  :  sin  zn_l  =  /x^ 
These  equations  being  multiplied  give 


Including  all  the  strata  of  the  atmosphere,  we  have 
zQ,  the  true  zenith  distance  ; 
zn,  the  apparent  zenith  distance  ; 

Mo  —  1  ; 

/mn,  the  index  of  refraction  at  the  station. 

Thus,  in  the  case  of  plane  strata,  the  relation  between  the  true 
and  the  apparent  zenith  distance  is  given  by  the  equation 

8mz0  =  jjLnsmzn  ............................  (23) 

This  equation  expresses  the  direction  of  the  ray  which,  strik- 
ing the  atmosphere  at  the  angle  zQt  is  refracted  so  as  to  reach 
the  earth  at  the  angle  zn.  Being  independent  of  all  the  values 
of  JUL  except  the  last,  it  proves  the  theorem. 

From  this  equation  we  may  derive  an  approximate  expression 
for  the  refraction  when  quantities  of  the  second  order  as  to  its 
amount  are  neglected.  Let  us  put  : 

R,  the  total  refraction. 
We  then  have  ZQ  =  zn  -f  E, 

and  developing  sin  00  to  quantities  of  the  first  order  in  R, 

sin  ZQ  =  sin  zn  •+•  R  cos  zn  .....................  (24) 


192  ASTRONOMICAL  REFRACTION  [§  103. 

Equating  (23)  and  (24),  we  have 

R  cos  zn  =  (u,  —  1)  sin  zn  } 

a      v*n        /  »  I  ,9-v 

£  =  (^-l)tan*nj- 

We,  therefore,  have  the  theorem : 

At  small  zenith  distances  the  refraction  is  proportional  to 
the  tangent  of  the  zenith  distance. 

Let  us  form  an  idea  of  the  error  of  this  theorem  for  a  zenith 
distance  of  45°.  From  what  has  already  been  shown  as  to  the 
density  of  the  atmosphere,  the  larger  portion  of  the  refraction 
takes  place  at  altitudes  below  15  kilometres.  The  supposed  ray 
reaches  this  height  at  the  horizontal  distance  of  1 5  km.  from  the 
station.  The  curvature  of  the  strata  within  this  distance  is 
approximately  8'.  The  numerical  value  of  /m  —  1  is,  as  we  shall 
see,  not  far  from  O0003.  The  change  in  the  refraction  produced 
by  a  change  of  8'  in  the  value  of  z  is  of  the  order  of  magnitude 
8'  X  0-0003;  or  about  0"15.  It  follows  that  the  correction 
required  by  the  law  of  tangents  is  small  even  at  a  zenith  distance 
as  great  as  45°. 

The  value  of  ^  —  1  at  ordinary  temperatures,  when  reduced  to 
arc,  ranges  between  57"  and  60".  The  number  of  degrees  in  the 
unit  radius  being  57'7,  it  follows  from  the  equation  (25)  that,  in 
the  immediate  neighbourhood  of  the  zenith  the  refraction  is 
approximately  1"  for  each  degree  of  zenith  distance. 

104.  Let  us  now  investigate  the  amount  of  the  refraction, 
supposing  the  atmosphere  to  consist  of  an  indefinite  number  of 
infinitely  thin  curved  strata,  each  of  the  thickness  dh. 


FIG.  18. 


Let  P  be  the  point  at  which  the  ray  intersects  the  bounding 
surface  S  between  any  two  consecutive  strata,  A  and  B ; 
f+df,  the  angle  of  incidence  at  P ; 
f,  the  angle  of  refraction  after  passing  P. 


§  105.]      REFRACTION  AT  SMALL   ZENITH   DISTANCES  193 

d£  is  then  the  change  of  direction  which  the  ray  undergoes 
from  one  stratum  into  the  next,  when  referred  to  any  fixed 
direction.  Putting  /m  and  /uL  +  d/m  for  the  indices  of  the  two 
strata,  we  shall  have 


whence  df=-tanf.  ..........................  (26) 

When  the  ray  reaches  the  lower  surface  of  stratum  B  at  a 
point  which  we  shall  call  P',  it  strikes  it  at  an  angle  of  incidence 
f-Mfi,  6^  being  the  angle  between  the  verticals  at  the  points 
P  and  P  '  .  It  follows  that  the  entire  change  of  f  from  stratum 
to  stratum  is  dg+  8^  ;  but  the  curvature  of  the  ray  being 
determined  by  reference  to  a  fixed  direction,  is  expressed  by  d£ 
alone.  The  preceding  equation  is,  therefore,  a  differential  equa- 
tion for  the  actual  amount  of  the  refraction. 

105.  Relation  of  density  to  refractive  index. 

To  express  (26)  in  terms  of  known  quantities,  we  have  next  to 
express  the  index  of  refraction  as  a  function  of  the  density  of 
the  air.  It  has  been  a  commonly  accepted  law  of  physics  that 
the  index  of  refraction  of  a  gas  is  given  as  a  function  of  its 
density  by  an  equation  of  the  form 

M2-1  =  2C/0,  ......  .  ....................  (27) 

c  being  a  constant  depending  on  the  wave-length  of  the  light 
and  p  the  density  of  the  gas.*  The  numerical  value  of  the 
constant  c  may  be  determined  either  from  the  observed  refraction 
of  the  stars,  or  from  laboratory  experiments.  In  practice  the 
astronomer  is  obliged  to  leave  out  of  consideration  the  variations 
in  the  refraction  of  light  with  its  colour,  and  to  base  all  his 
computations  on  the  hypothesis  that  there  is  a  certain  mean  or 
brightest  region  in  the  visible  spectrum,  the  light  of  which  he 
alone  observes.  There  is  an  unavoidable  indetermination  as  to 
the  choice  of  a  particular  ray  of  the  spectrum  for  this  purpose, 


*  Recent  experimental  investigations  make  it  probable  that  the  yet  simpler  law 
/i=l+c/>  may  be  as  near  or  nearer  to  the  truth  ;  but,  for  astronomical  purposes, 
the  difference  between  the  results  of  the  two  formulae  is  unimportant. 
N.S.A.  N 


194  ASTRONOMICAL  REFRACTION  [§  105. 

added  to  which  the  differences  between  the  colours  of  the 
different  stars  must  involve  differences  in  the  refraction  of  their 
light.  It  is,  however,  remarkable  that  the  most  careful  observa- 
tions made  up  to  the  present  time  do  not  show  any  differences 
arising  from  this  cause  at  moderate  altitudes. 

But  this  cause  assumes  another  form  near  the  horizon,  where 
its  effect  must  be  sensible.  Here  the  blue  and  even  the  green 
rays  are  nearly  all  absorbed  by  the  atmosphere,  leaving  the 
visible  body  to  be  represented  by  rays  from  the  lower  part  of 
the  spectrum.  The  refraction  must  then  be  somewhat  less  than 
it  would  be  as  determined  by  any  law  which  assumed  the  index 
of  refraction  to  be  the  same  at  all  altitudes. 

Owing  to  this  indetermination  it  is  better  to  determine  the 
index  of  refraction,  or  the  value  of  c,  from  astronomical  observa- 
tions than  from  laboratory  measures.  The  two  are,  however,  in 
so  good  agreement  that  it  is  indifferent  which  we  accept  as  the 
numerical  basis  of  a  theory. 

Added  to  the  source  of  indetermination  just  mentioned,  we 
.have  the  fact  that  the  best  investigations  of  the  actual  refraction 
suffered  by  the  stars  show  a  range  of  the  thousandth  part.  In 
fact,  Bessel's  refraction  tables,  which  have  not  yet  gone  wholly 
out  of  use,  are  based  on  an  index  of  refraction  greater  by  0*003 
of  its  amount  than  that  of  the  Poulkova  tables,  which  were 
constructed  from  the  most  refined  observations.  Yet  the  general 
consensus  of  recent  observations  is  toward  the  view  that  the 
constant  of  the  latter  tables  still  requires  a  diminution  of  perhaps 
its  thousandth  part. 

In  the  numerical  investigations  of  the  present  chapter  the 
adopted  value  of  c  is  c  _  Q-226  07 

a  value  which  agrees  closely  with  the  Poulkova  tables,  and  with 
the  best  laboratory  measures  of  the  refractive  power  of  light 
near  the  ray  D. 

106.  Form  in  which  the  refraction  is  expressed. 
By  differentiating  (27),  we  find 

Ijidfji  =  cdp, 


§  106.]    FORM  IN   WHICH   REFRACTION   IS   EXPRESSED          195 

These  give,  by  division, 

d/m  _    cdp 
~ 


The  density  of  the  air  is  a  minute  fraction  ranging  from  0  to 
•0012  or  '0013,  the  usual  value  of  p  at  the  earth's  surface.  We 
put  plt  the  value  of  p  at  the  point  of  observation.  As  p  is  a 
small  factor,  ranging  from  the  value  0  at  the  upper  limit  of  the 
atmosphere  to  p^  at  the  station,  we  may,  without  appreciable 
error,  replace  the  divisor  1  -\-2cp,  which  ranges  between  1  and 
T0006,  by  its  mean  value,  1  +cpv  which  will  give 


A  /D! 

Using  the  notation 


and  putting  R  for  the  amount  of  the  refraction,  (26)  will  give 
for  its  differential 

This  equation  is  rigorous.  Conceive  that  we  integrate  it  through 
the  course  of  the  ray  from  the  station  of  the  observer,  where  we 
have  £=  apparent  zenith  distance  =  z, 

to  the  limit  of  the  atmosphere  where 

The  total  refraction  will  then  be  equal  to  the  definite  integral 

c     tanfdp '-(31) 


Jo 

Since  £  does  not  differ  greatly  from  z  at  moderate  zenith 
distances,  it  follows  that  a  first  approximation  to  the  refraction 
in  the  region  around  the  zenith  will  be  derived  by  integrating 
as  if  tan  f  had  the  constant  value  z,  thus  giving 

R  =  cpl  tan  z, 

a  result  which,  in  principle,  is  equivalent  to  that  expressed  by 
(25).  It  follows  that  if  we  determine  a  factor  m  by  the  con- 
dition that  the  refraction  shall  be  given  rigorously  in  the  form 

R  =  mCp-L  tan  z,   (32) 


196  ASTRONOMICAL  REFRACTION  [§  106. 

this  factor  will  differ  little  from  unity  at  moderate  zenith 
distances.  Its  investigation  being  somewhat  more  intricate  than 
is  appropriate  to  the  present  section,  is  deferred  to  the  following 
one.  At  present  we  shall  show  how,  m  being  taken  as  known, 
the  refraction  is  practically  determined. 

107.  Practical  determination  of  the  refraction. 

The  density  of  the  air  at  the  station,  or  pv  being  the  unknown 
factor  in  (32),  we  have  to  begin  by  showing  the  practical  method 
of  determining  it  at  the  moment  of  observation,  and  of  bringing 
it  into  the  theory.  Its  value  is  determined  primarily  by  Eq.  (2), 
§94, 

/°=      • 
yr 

Here  y  is  an  absolute  constant  ;  p  can,  therefore,  be  immediately 
determined  when  the  pressure  p  and  the  temperature  r  are 
known.  These  two  quantities  are  given  by  the  readings  of  the 
barometer  and  thermometer.  If  the  thermometer  reads  t  degrees 
centigrade,  the  expression  for  r  is 

T  =  27l-5  +  <  ............................  (33) 

If  the  scale  is  that  of  Fahrenheit,  the  expression  is 


The  value  of  T  being  found,  we  have  shown  in  §  95,  Eq.  (9), 
that  the  value  of  p  is  given  by  the  equation 


(34) 


where  b  is  the  ratio  of  the  observed  to  the  standard  height  of 
the  barometer,  the  former  being  corrected  for  the  temperature 
of  the  mercury  in  the  barometer,  while  g  is  the  ratio  of  gravity 
at  the  place  to  gravity  at  Paris. 
Let  us  put 

B,  the  observed  reading  of  the  barometer  ; 

B0,  the  standard  height,  760  mm.; 

AC,  the  coefficient  of  cubical  expansion  of  mercury  for  1°; 

tf,  the  temperature  of  the  mercury  above  0°  C. 
In  strictness  we  should  take  for  AC  the  excess  of  the  cubical 
expansion  of  mercury  over  the  cubical  expansion  of  the  tube  of 


§  107.]    PRACTICAL   DETERMINATION  OF  REFRACTION          197 

the  barometer.  But  the  latter  is  so  small  that  it  is  neglected  in 
practice,  although  there  is  no  difficulty  in  the  introduction  of  its 
approximate  value  for  the  special  substance  of  which  the  tube  is 
composed. 

With  these  data  the  value  of  6  will  be  given  by  the  equation 


Introducing  these  various  quantities,  and  putting 

6r,  the  force  of  gravity  at  the  place,  in  any  measure  whatever  ; 

6r0,  the  intensity  of  gravity  at  Paris,  in  the  same  measure; 

the  expression  for  pl  is  given  in  terms  of  known  and  observed 

quantities  by  the  equation 

_  0-351  11     0  B 

**  T  'flo'a+KW 

Substituting  this  value  of  p1  in  (32)  the  refraction  will  be 
expressed  as  a  product  of  the  five  factors  : 

0-351  11     B         1         G 
cmtanz;       —  -;        ;     -;       ..........  (36a) 


The  logarithms  of  the  first  four  of  these  factors  are  tabulated 
in  different  refraction  tables  used  by  the  observatories.  The 
factor  depending  on  gravity  has  heretofore  been  very  generally 
neglected,  but  should  always  be  introduced  if  the  fundamental 
constant  c  is  not  determined  at  the  observatory  itself.  The 
first  term  is  tabulated  as  a  function  of  0,  the  apparent  zenith 
distance;  the  others  as  functions  of  the  actual  reading  of  the 
external  thermometer  and  barometer  and  the  attached  ther- 
mometer, which  gives  the  temperature  of  the  mercury.  As  to 
the  special  scale  of  temperature  and  of  barometer  height  to  be 
used  as  the  argument  of  the  tables,  it  need  only  be  remarked 
that  attention  should  be  paid  to  see  that  the  particular  scale  for 
which  the  tables  are  constructed  is  that  to  which  the  instruments 
are  graduated. 

The  factor  m,  depending  as  it  does  on  the  curvature  of  the 
strata,  differs  little  from  unity  in  the  neighbourhood  of  the 
zenith.  A  little  consideration  will  make  it  evident  that  m  must 
diminish  as  the  zenith  distance  increases,  because  the  angle 


198  ASTRONOMICAL  REFRACTION  [§  107. 

between  the  course  of  the  ray  and  the  vertical  at  any  point  on 
the  ray  constantly  diminishes  with  increasing  height  owing  to 
the  curvature  of  the  strata.  Since  tan  z  increases  without  limit 
as  the  horizon  is  approached,  while  the  refraction  remains  finite, 
the  factor  m  must  vanish  at  that  point,  and  its  logarithm  must 
become  infinite.  The  latter  cannot,  therefore,  be  advantageously 
used  near  the  horizon. 

The  determination  of  m  is  necessary  to  the  completeness  of 
the  theory.  But  as  its  value  depends  upon  the  law  of  diminution 
of  density  with  increasing  height,  which,  as  has  been  seen  in  the 
preceding  section,  is  very  largely  hypothetical,  there  can  be  no 
easily  defined  theory  for  the  determination  of  m.  As,  on  the 
hypotheses  which  come  nearest  to  the  truth,  the  developments 
necessary  to  determine  this  factor  become  intricate,  the  discussion 
of  the  subject  is  deferred  to  the  following  section. 

108.  Curvature  of  a  refracted  ray. 

Let  us  now  investigate  the  radius  of  curvature  of  the  refracted 
ray  when  near  the  surface  of  the  earth.  We  put 

r,  the  radius  of  curvature  ; 

s,  the  length  measured  along  the  ray. 
The  radius  of  curvature  is  given  by  the  differential  equation 

l=d£ 

T~  ds' 

ds  being  the  element  of  length  of  the  ray. 

The  algebraic  sign  which  we  assign  to  d£  is  indifferent  ;  we 
shall  therefore  always  regard  both  dg  and  R  as  positive  or 
signless  quantities. 

The  value  of  d£  is  given  by  (26).  Substituting  for  d/u.  :  fj.  its 
value  (28),  we  have 


1+Cft 

The  product  Cp1  in  the  denominator  is  so  small  that  for  our 
immediate  purpose  it  may  be  disregarded.  Substituting  for  dp 
its  value  (7),  we  have 


§  108.]  CURVATURE   OF  A  REFRACTED  RAY  199 

The  element  of  the  length  of  the  ray  contained  in  the  stratum 
is  ds  =  dh  sec  f. 

The  quotient  of  these  equations  give 

dt     1          /I   ,  1  dr\   .  /0*x 

-*»  =  -  =  cp   —H 77    smz,  (37) 

ds     r      ^V/^     r  dh,/ 

where  we  introduce  the  reciprocal  of  the  pressure-height  for  — . 

This  equation  shows  that  the  curvature  of  the  ray  varies 
directly  as  sin  z,  and,  therefore,  has  its  maximum  value  when  the 
ray  is  horizontal.  Let  us  next  compute  the  value  of  the  curva- 
ture for  this  case.  We  shall  begin  by  supposing  the  temperature 
to  be  uniform.  The  equation  (37)  then  gives  for  the  radius  of 
curvature 

r  =  -x (38) 

cp 

Taking  the  case  of  air  at  standard  density,  we  have,  from 
numbers  already  given, 

Cpo  =  0-000  293, 
and  putting  7^  =  8  km., 

r  =  27  300  km.  =  4'3  radii  of  the  earth. 

We  have  already  learned  that,  near  the  earth's  surface,  the 
rate  of  change  of  temperature  with  height  is  very  variable. 
Taking  —  6°'5  C.  per  kilometre  as  a  normal  rate,  and  10°  C.  as  a 
normal  temperature,  we  shall  have 


,(39) 


5?=  0-0231, 

r  dri 

Cp  =  0-000  2821, 
^  =  8-28  km., 
r  =  34600  km.,  =  5'44  radii  of  the  earth. J 

As  during  the  day  the  rate  of  diminution  is  yet  greater  than 
6°'5  per  km.,  we  may  regard  the  ordinary  curvature  of  a  nearly 
horizontal  ray  as  ^  that  of  the  geoid.  But,  owing  to  the  cause 
already  mentioned,  this  number  is  subject  to  wide  variations. 
Not  unfrequently  the  temperature-gradient  along  the  course 


200  ASTKONOMICAL   REFRACTION  [§  108. 

of  the  ray  is  positive.  This  is  nearly  always  the  case  within  an 
observing  room,  and  within  the  tube  of  a  telescope,  owing  to  the 
tendency  of  the  warmer  air  to  rise  to  the  top  of  a  room  or  to 
the  highest  part  of  a  tube.  In  this  case  the  curvature  will  be 
greater  than  the  normal.  This  particular  phase  of  the  subject 
belongs  to  the  field  of  practical  and  instrumental  astronomy,  and 
will  not  be  further  considered  at  present. 

The  same  result  follows  when  a  body  of  warm  air  passes 
over  the  frozen  surface  of  the  Arctic  seas.  At  a  certain 
temperature-gradient  the  curvature  of  the  ray  may  become  equal 
to  or  greater  than  that  of  the  ocean  itself.  Then  there  will  be 
no  limit  to  the  distance  at  which  objects  may  be  seen  except  that 
arising  from  non-transparency  of  the  air.  It  is  easy  to  define 
the  temperature -gradient  at  which  this  effect  follows.  We  have 
only  to  insert  for  r  in  (37)  the  value  of  the  earth's  radius,  put 

sin  0  =  1,  and  thus  determine  the  value  of  -77-  as  an   unknown 

ati 

quantity.     We  then  have 

^  =  -^---=117°,  .  (40) 

ah,     acp     y 

where  a  is  the  radius  of  curvature  of  the  geoid. 

This  implies  a  diminution  of  a  little  more  than  1°  C.  or  nearly 
2°  F.  for  each  10  metres  of  height. 

The  contrary  case  arises  when  the  surface  of  a  flat  plain  is 
heated  by  the  rays  of  the  sun.  If  the  temperature  gradient 
then  has,  near  the  ground,  a  negative  value  exceeding  1°  in 
34  metres,  the  ray  will  be  concave  toward  the  earth.  For 
negative  gradients  largely  exceeding  this  limit  the  ground  at  a 
distance  may  not  be  visible  at  all,  a  ray  the  line  of  which  would 
reach  the  ground  at  a  small  angle  being  bent  upwards  and  thus 
showing  to  the  observer  only  the  sky,  while  a  higher  ray  where 
the  gradient  is  more  nearly  normal  will  pass  to  and  show  an 
elevated  object  at  a  distance.  In  cases  of  this  sort  with  a 
positive  and  rapidly  diminishing  gradient,  an  inverted  image  of 
distant  objects  may  be  seen.  It  is  to  this  action  of  the  tem- 
perature-gradient that  the  varied  phenomena  of  mirage  are  due. 


§  109.]       DISTANCE  AND   DIP  OF  THE   SEA  HORIZON 


201 


109.  Distance  and  dip  of  the  sea  horizon. 

By  the  sea  horizon  is  meant  the  apparent  boundary  of  the 
surface  of  the  ocean  when  viewed  through  a  transparent 
atmosphere.  The  plane  of  the  observer's  horizon  is  necessarily 
above  the  ocean,  the  latter  receding  farther  and  farther  below  it 
at  a  greater  and  greater  distance.  Consequently,  the  sea  horizon 
is  depressed  below  this  plane.  The  angular  amount  of  this 
depression  is  called  the  dip  of  the  horizon. 

In  Fig.  19  let  the  arc  HS  be  a  section  of  the  ocean  surface 
through  the  centre  of  the  earth  C ; 


FIG.  19. 

E,  the  position  of  the  observer's  eye ; 

S,  the  point  in  which  the  observer's  vertical  line  intersects  the 
surface  of  the  ocean; 

EL,  a  section  of  the  horizontal  plane  through  E; 

EK,  a  tangent  from  the  eye  of  the  observer  to  the  ocean 
surface  at  K. 

Were  there  no  refraction  this  tangent  would  be  the  course  of 
a  ray  passing  .  between  K  and  the  observer.  Since  LEG  and 
GKE  are  both  right  angles,  it  follows  that  the  angle  LEK,  or 
the  dip  of  the  horizon,  would  then  equal  EGK.  That  is  to  say, 
the  dip,  expressed  in  minutes,  would  be  equal  to  the  distance 
of  the  horizon  in  nautical  miles.  But,  owing  to  the  effect  of 
refraction,  the  actual  ray  EH  is  concave  to  the  surface  of  the 
ocean ;  consequently  it  is  tangent  to  the  latter  at  a  point  more 
distant  than  K.  Let  H  be  this  point.  We  see  from  the  figure 
that  the  actual  dip  is  less  than  the  geometric  dip,  and  the  actual 


202  ASTRONOMICAL   REFRACTION  [§  109 

distance   of  the  sea   horizon  greater  than  the  distance  of  the 
geometrical  horizon. 

To  investigate  the  actual  distance  and  dip,  we  remark  that  the 
height  of  the  observer's  eye  above  the  ocean  SE  is  practically  so 
small  that  we  may  treat  the  ray  as  horizontal  and  disregard  the 
effects  of  the  height,  thus  supposing  the  curvature  of  the  ray 
uniform.  Let  a  point  D,  not  shown  in  the  figure,  be  the  centre 
of  curvature  of  the  ray  HE,  which  lies  on  the  line  HG  produced. 
In  the  triangle  CDS  we  therefore  have 

Side  DC  =  r  —  a,  r  being  the  radius  of  the  curvature  of  the  ray. 
We  also  put  Side  DS  =  s, 

Angle  HC8=C, 
so  that  the  angle  at  C  of  the  triangle  DSC  is  180°  -C. 

From  a  well-known  theorem  of  geometry  we  have 

DS2  =  DC'2  +  CS2  +  WC  .  CS  cos  (7, 
which  gives  the  equations 


4a(r  —  a) 

It  will  be  seen  that  C  is  the  distance  of  the  sea  horizon  in 
arc  at  the  earth's  centre.  This,  when  expressed  in  minutes, 
corresponds  to  nautical  miles.  Let  D  be  the  distance  of  the 
horizon  expressed  in  this  way;  then,  owing  to  the  minuteness 
of  C,  we  may  put  J)  =  6876  sin  \  C, 

the  number  6876  being  twice  the  number  of  minutes   in  the 
radian. 

Owing  to  the  minuteness  of  the  height  HE  of  the  observer's 
eye  above  the  ocean,  and  of  the  angle  C,  we  may  treat  these 
quantities  as  infinitesimals.  Putting 

h,  the  height  of  the  eye, 

we  shall  then  have,  with  all  required  precision, 

r  —  s  =  h, 


sin 


Va(r-a)' 


§  110.]         DISTANCE  AND  DIP  OF  THE  SEA  HORIZON  203 

With  these  various  substitutions  we  find  that  the  distance  D 
is  given  in  nautical  miles  by  the  equation 


B^attW^U (42) 

V  a(r  —  a) 

The  value  of  D  therefore  depends  upon  the  radius  of  curvature 
r  of  the  ray,  which  again  is  a  function  of  the  temperature- 
gradient.  We  have  already  shown  how  to  express  r  as  a 
function  of  this  gradient.  For  all  ordinary  practical  purposes 
we  may  suppose  r  —  Qa. 

Then,   taking    the    metre    as   the   unit   of    length,    we   have 
and 

/T7TE 

(43) 


For  the  dip  of  the  horizon  may  be  taken  the  angle  8  of  the 
triangle  CSD,  which  will  give,  with  all  required  precision, 


Dip  =  ?     -sin  (7=^- 


r  v        ar 

When  r  =  6a  this  gives 


Dip  in  minutes  =  3438 A/ ^- 


We  conclude  that  the  distance  of  the  sea  horizon  in  nautical 
miles  is  about  y  greater  than  the  square  root  of  the  height  of 
the  observer's  eye  in  feet.  From  the  deck  of  an  ocean  liner,  on 
which  the  eye  is  about  20  feet  above  the  sea,  the  distance  is  not 
far  from  5  miles.  The  line  of  sight  being  tangent  to  the  ocean, 
its  height  above  the  ocean  beyond  the  horizon  is  given  by 
the  reversal  of  the  formula  for  the  distance.  From  this  it 
may  be  concluded  that,  at  a  distance  of  10  miles,  only  the  bridge 
of  another  ship  will  be  visible,  and  that  15  or  16  miles  is  about 
the  greatest  distance  at  which  the  upper  parts  of  the  smokestack 
can  be  seen. 

Section  III.    General  Investigation  of  Astronomical 
Refraction. 

110.  In  the  preceding  section  the  results  of  refraction  have 
been  treated  rigorously  only  as  regards  the  fundamental  principles 


204  ASTRONOMICAL  REFRACTION  [§110. 

and  on  the  hypothesis  that  the  equiponderant  strata  of  the 
atmosphere  are  plane.  We  have  now  to  consider  more  rigor- 
ously than  before  the  effect  of  the  curvature  of  the  strata 
upon  the  refraction.  In  strictness  the  equiponderant  surfaces 
should  be  regarded  as  ellipsoidal,  corresponding  approximately 
to  the  figure  of  the  geoid.  It  follows  that  in  the  region  of  the 
two  poles,  the  curvature  of  the  surfaces  may  be  regarded  as 
spherical,  while,  at  the  equator,  the  differences  in  the  different 
azimuths  reaches  its  maximum.  The  question  whether  this 


FIG.  20. 

difference  of  curvature  in  different  azimuths  and  different 
latitudes  appreciably  affects  the  refraction  can  only  be  deter- 
mined after  the  formulae  for  the  amount  of  refraction  have  been 
developed  and  reduced  to  numbers.  It  may,  however,  be  re- 
marked that,  since  the  observations  for  which  a  precise  value 
of  the  refraction  is  necessary  are  almost  exclusively  those 
made  in  the  meridian,  the  main  question  will  be  whether  the 
differences  in  the  curvature  of  the  meridian  in  different  latitudes 
have  a  sensible  effect. 


§110.]    INVESTIGATION  OF  ASTEONOMICAL  REFRACTION     205 

In  Fig.  20,  let 

0  be  the  centre  of  curvature  of  the  geoid  in  that  plane  passing 
through  the  station  with  reference  to  which  the  refraction  is  to 
be  expressed ; 

8,  the  station  of  observation  ; 

ES,  the  level  surface  of  the  station. 

Ordinarily  the  station  will  be  situated  near  the  surface  of  the 
geoid,  so  that  ES  may  be  regarded  as  representing  this  surface, 
but  in  theory  we  may  regard  S  as  at  any  elevation  above  it. 

Z,  the  zenith  of  the  station; 

ZQ,  the  upper  surface  of  a  thin  stratum  A  within  which  we 
regard  the  index  of  refraction  as  constant ; 

PQR8,  the  course  of  a  ray  of  light  undergoing  refraction  at 
the  points  Q  and  R  on  the  upper  and  lower  surfaces  of  the 
stratum  A,  and  ultimately  reaching  S\ 

OQ,  ORW,  vertical  lines  from  the  centre  of  curvature  through 
the  points  Q  and  R. 

We  assume  that  the  level  surfaces  are  all  concentric  and 
therefore  equidistant. 

In  order  to  express  the  refraction  we  begin  with  the  con- 
ception of  the  atmosphere  as  formed  by  an  indefinite  number  of 
successive  strata,  each  of  uniform  density.  Then  by  passing  to 
the  limit  in  which  the  number  of  strata  becomes  infinite  and  the 
differences  of  density  in  two  consecutive  strata  infinitesimal, 
we  have  the  case  of  the  continuously  varying  density  of  the 
atmosphere. 

Let  us  now  study  the  refraction  which  a  ray  suffers  in  passing 
from  P  through  Q  and  R.  For  this  purpose  we  put 

//,,  the  index  of  refraction  for  the  stratum  next  above  A  whose 
lower  surface  is  Q  WZ ; 

p,  the  index  for  the  stratum  A  itself ; 

£  the  angle  of  incidence  at  which  the  ray  falls  at  Q  upon  the 
upper  surface  of  the  stratum  A  ; 

f ',  the  angle  of  refraction  OQR  at  which  it  enters  the  stratum  A] 

£p  the  angle  of  incidence  QRW  at  which  it  strikes  the  lower 
surface  of  A  ; 

r,  r.p  the  radii  of  curvature  of  the  upper  and  lower  surfaces  of 
the  stratum  A. 


206  ASTRONOMICAL  REFRACTION  [§110. 

The  law  of  refraction  gives  the  equation 

/of  sin  f '  =  /UL  sin  f . 
In  the  triangle  ROQ  we  have 

Angle  Q  =  f, 
Angle  ,R=  180° -&, 
whence  r  sin  f '  =  ?^  sin  fr 

Eliminating  f '  from  these  equations,  we  have 
r^j!  sin  f  x  =  r/x  sin  f . 

It  follows  that  the  product  r/x  sin  f  has  the  same  value  in 
every  two  consecutive  strata,  and  is  therefore  constant  for  the 
whole  course  of  the  ray  through  the  atmosphere.  Its  value 
may,  therefore,  be  derived  from  its  value  at  the  base,  to  express 
which  we  put 

a,  the  radius  of  curvature  of  the  geoid  at  the  station  : 
JULV  the  index  of  refraction  at  that  point ; 
z,  the  apparent  zenith  distance  of  the  body. 
The  value  of  the  constant  in  question  will  thus  be 

etytj  sin  z=C. 

Passing  now  to  the  actual  case  in  which  the  increase  of 
density  is  a  continuously  varying  quantity,  the  successive  strata 
become  infinitely  thin,  and  the  angle  f  becomes  that  which  the 
ray,  at  each  point  of  its  course,  makes  with  the  vertical  line  at 
that  point.  We  therefore  have  the  equation 

rjm  sin  f  =  a/xx  sin  z (1) 

The  second  member  of  this  equation  being  supposed  to  contain 
only  given  and  known  quantities,  the  equation  expresses  a 
relation  between  r,  /x,  and  f  at  all  points  on  the  course  of  any 
one  fay. 

The  refraction  being  the  total  change  which  the  direction  of 
the  ray  undergoes  from  the  point  of  observation  to  the  outer 
limit  of  the  atmosphere,  it  follows  that  the  differential  of  the 
refraction  is,  at  each  point,  the  infinitesimal  curvature  of  the 
ray.  This  is  the  same  as  that  part  of  the  change  in  the  angle  f 
which  arises  from  the  change  in  the  value  of  the  index  /m  from 


§  111.]  INVESTIGATION  OF  ASTRONOMICAL  REFRACTION      207 

one  point  to  another.      We   therefore  have  the  differential  of 
the  refraction  by  writing  dR  for  d£  in  equation  (26),  Sect.  II., 

c^=-°^tan£ (2) 

JUL 

In  this   equation    we    substitute   for   tan  f  its  value  derived 

from  W> 

Q^~M-Mlasnz 
The  differential  equation  (2)  of  the  refraction  thus  becomes, 
dropping  the  negative  sign  as  indifferent, 

77?     dp.  sinz  Sy. 

' 


This  is  a  rigorous  equation  which,  being  integrated  as  to  the 
variable  //.  from  the  outer  limit  of  the  atmosphere  to  the  point 
of  observation,  or  the  reverse,  gives  the  total  refraction. 

111.  Transformation  of  the  differential  equation. 

In  the  integration  the  apparent  zenith  distance  z,  and  the 
radius  of  curvature  a,  are  constants,  while  /x  and  r  are  variables. 
To  reduce  the  expression  to  an  integrable  form,  a  number  of 
transformations  are  required. 

Our  first  two  transformations  will  consist  in  replacing  -  by  its 

a 

expression  in  terms  of  the  height,  and  expressing  —  in  terms  of 
the  density  of  the  air.     Putting,  as  before,  h  for  the  height, 

=  1  +  -.  ...(4) 

a  a 

It  will  conduce  to  clearness  to  express  h  in  terms  of  the 
pressure  height  \  as  the  unit  of  height.  This  we  do  by  intro- 
ducing the  variable  x,  -L 


We  have  found  h^  to  be  a  function  of  the  temperature  rx  at  the 
station  defined  by  the  equation 


208  ASTRONOMICAL   REFRACTION  [§111. 

We  then  have  -  =  1  +  ^  x. 

a  a 

From   the  relation  between  the  index  of  refraction  and  the 

..............................  (7) 


we  have 


By  introducing  the  constant  a, 


and  the  variable  w,  w  =  1  —  —  ,    ...............................  (9) 

Pi 

we  have  —»  =  1  —  2a.w. 

Mi- 
Then,  putting  for  brevity, 


a 


7* 

which  gives  -  =  1  +  vx, 

Cb 

.,2     ^2 

we  have  -^  -^=l  +  2u,  ..............................  (11) 


where  u  =  vx  —  OLW+^vV  —  oivwx(2  +  vx)  ................  (12) 

Making  in  the  equation  (3)  the  substitutions  (11)  and  (28)  of 

§  106,  we  have  c  d 

(12a) 


Our  next  step  is  to  substitute  w  for  p  as  the  variable.     From 
(9),  dp=-pldw  ............................  (13) 

Substituting  for  dp  this  value,  we  shall  put 


(14) 


With  these  substitutions  the  general  equation  for  the  refraction 
may  be  written  in  either  of  the  forms, 

dw  (15o) 


p 

R  =  &  sin  z  I   — 7— 

Jov/2 


§  112.]  TRANSFORMATION  OF  DIFFERENTIAL  EQUATION       209 

The  first  of  these  forms  is  most  conveniently  applicable,  except 
within  a  few  degrees  of  the  horizon,  when  we  are  obliged  to  have 
recourse  to  the  second.  It  was  shown  in  Section  II.  that  for  small 
zenith  distances  the  refraction  is  proportional  to  the  tangent  of 
the  zenith  distance,  and  that,  therefore,  except  near  the  horizon, 
the  refraction  is  usually  expressed  in  the  form 

R  =  am  tan  z (16) 

We  now  see  that  the  factor  m  will  be  given  by  the  equation 


the  investigation  of  which  next  demands  our  attention. 

112.  The  integration. 

The  great  problem  of  astronomical  refraction  consists  in  the 
integration  of  the  preceding  equations.  This  problem  offers  no 
serious  difficulty  in  the  case  of  zenith  distances  to  80°,  except 
that  of  deciding  upon  a  law  according  to  which  the  density  of 
the  air  diminishes  with  the  height.  It  will  be  well  to  preface  a 
consideration  of  the  problem  by  a  review  of  its  general  nature, 
and  of  the  forms  in  which  it  has  to  be  attacked  in  different 
cases.  First  let  us  form  an  idea  of  the  order  of  magnitude 
of  the  quantities  with  which  we  have  to  deal,  especially  of  u. 

From  what  we  have  seen  of  the  law  of  density  of  the  air,  it 
follows  that  all  the  refraction  with  which  we  need  concern  our- 
selves takes  place  below  the  altitude  of  60  kilometres,  and  that 
it  is  very  small  above  40  kilometres.  For  this  altitude  we 
have,  approximately,  _  „ 


AT 
Also 


a     6360     800 
Hence  we  always  have 


while  values  greater  than  0'006  add  very  little  to  the  refraction. 
w  ranges  between  the  limits  0  and  1,  and  as  oc  <  0*0003,  it 
follows  that  aw  <  0.0003 

N.S.A.  0 


210  ASTRONOMICAL  EEFRACTION  [§  112. 

These  are  the  two  largest  terms  of  u,  so  that,  whenever 

sec0<  7, 

we  have  2u  sec2  z  <  1, 

and  the  denominator  of  (17)  may  be  developed  in  powers  of 
this  quantity. 

Regarding  v  and  oc  as  quantities  of  the  first  order,  the  three 
last  terms  of  u  in  (11)  are  of  the  second  and  third  orders. 
Owing  to  the  minuteness  of  v  and  oc  these  higher  terms  are  of 
minor  importance  and  their  consideration  will,  therefore,  be 
postponed.  Dropping  them  from  the  value  of  u,  the  latter 
becomes 

u  =  vx  —  OLW (18) 

The  difficulty  of  the  problem  arises  from  the  fact  that  the  two 
terms  of  u  must  be  expressed  as  a  function  of  some  one  variable 
before  the  integration  can  be  effected.  Between  these  terms  the 
relation  is  of  a  complex  character.  It  will  be  profitable  to  appre- 
hend the  respective  origins  of  the  two  terms  of  u.  We  recall 
that  x  may  be  defined  as  the  altitude  above  the  earth  of  any 
point  of  the  refracted  ray,  expressed  in  terms  of  the  pressure 
height  as  the  unit.  The  factor  v  arises  from  the  rotundity  of 
the  earth,  being  the  reciprocal  of  the  earth's  radius  of  curvature, 
which  would  vanish  were  there  no  curvature.  The  product  vx 
may  be  described  as  due  to  the  change  of  the  angle  between  the 
ray  and  the  vertical  line  at  each  point,  so  far  as  this  change  arises 
from  the  earth's  rotundity.  Considered  as  passing  in  the  reverse 
direction  from  the  observer  outwards,  the  height  of  the  ray  at  any 
point  depends  both  on  the  curvature  of  the  earth  and  on  that  of 
the  ray  itself.  The  term  OLW  may  be  described  as  arising  from  the 
curvature  of  the  ray. 

Assuming  the  density  of  the  air  to  be  a  given  function  of  the 
height,  the  quantity  w, 

w  =  l  —  — 
Pi 

becomes  a  function  of  x,  which,  being  substituted  in  u,  enables 
the  latter  to  be  expressed  as  a  function  of  x.  Then,  replacing  the 
differential  of  w  by  that  of  x,  the  problem  will  become  that  of 


§113.]  THE   INTEGRATION  211 

integrating  with  respect  to  x  as  the  independent  variable.  This 
offers  no  difficulty  in  using  the  form  (15a),  but  is  not  practicable 
when  sec  z  is  so  large  that  the  form  (b)  has  to  be  used. 

In  Section  I.  five  hypotheses  have  been  set  forth  as  to  the 
relation  between  the  density  of  the  air  and  the  height.  These 
hypotheses  lead  to  as  many  expressions  for  the  relation  between 
w  and  x.  Of  these  hypotheses  it  may  be  said,  in  a  general 
way,  that  the  first  three  lead  to  forms  which  admit  of  being 
integrated  by  well-known  methods  ;  but  that  all  three  of  them 
deviate  in  a  definable  way  from  the  actual  facts  of  the  case. 
The  remaining  two,  by  adopting  the  proper  factor  of  diminution 
of  density  with  altitude,  can  probably  be  made  to  represent  the 
facts  as  accurately  as  is  necessary  for  the  purpose  of  refraction. 

113.  Development  of  the  refraction. 

The  preceding  considerations  suggest  the  separate  considera- 
tion of  the  problem  in  its  two  forms.  In  one  of  these  forms  we 
use  (15a)  and  develop  in  powers  of  u\  in  the  other  we  have 
to  consider  the  method  of  dealing  with  the  integration  when  the 
required  development  in  powers  of  u  is  impracticable.  The 
limits  of  the  zenith  distance  within  which  the  first  method  is 
applicable  will  appear  after  the  developments  are  effected. 

The  purpose  of  this  method  is  the  determination  of  m  from 
the  equation  (17).  The  development  of  the  denominator  in 
powers  of  u  by  the  binomial  theorem  gives 

(l  +  2usec2^)~*  =  l-usec20+fi62sec4^...,  ............  (19) 

the  coefficient  of  (  —  l)*u*  being 


The  first  five  coefficients,  taken  positively,  are 


[2]=    f 
[3]=    f 


.(20) 


212  ASTRONOMICAL  REFRACTION  [§  113. 

If  we  express  m  in  the  form 

m  =  1  —  mx  sec2z  +  m2  sec4z  —  . . .  ,  (21) 

the  general  value  of  mi  will  be 

<Zw (22) 


Putting  for  un  its  value  in  terms  of  ran,  the  definite  integrals 
which  enter  into  this  expression  are  found  by  integrating  the 
following  forms  between  the  limits  0  and  1 : 

u  dw  =  vx  div  —  caw  dw 

—  2oivxwdw+a} 

•(23) 


u3dw  = 


u5dw  = 


When  dw  is  replaced  by  its  value  in  terms  of  dx,  the  limits  of 
integration  will  be  0  and  the  value  of  x  corresponding  to  the 
height  of  the  atmosphere,  or  of  the  absolute  zero  of  temperature. 

When  taken  between  these  limits,  the  integrals  admit  of 
farther  simplification.  The  last  terms  in  each  of  the  differentials 
being  independent  of  xy  are  numerical  constants  multiplied  into 
successive  powers  of  a.  They  are.  therefore,  like  the  principal 
terms  of  the  refraction,  dependent  only  upon  the  density  of  the 

air  at  the  point  of  observation.     To  reduce  \xd/w,  we  substitute 
for  dw  its  expression  in  terms  of  dp, 

dw= -. 

Pi 

We  thus  have 


Jxdw= \xdp. 
PiJ 

Integrating  by  parts, 

\xdp  =  px—  \pdx. 


,co  l 

I   pdx=T 
Jor          A 


§  114.]  DEVELOPMENT  OF  THE  EEFBACTION  213 

At  the  lower  limit  of  integration  we  have  x  —  0,  and  at  the 
upper  limit  p  =  0.  Hence  the  product  px  vanishes  at  both 
limits,  leaving  as  the  integral 

i  r°° 
pdh. 

This  integral  expresses  the  total  mass  of  the  atmosphere 
contained  in  a  vertical  column  of  unit  base.  It  is,  therefore, 
independent  of  the  law  of  density.  We  therefore  have  the 
remarkable  theorem : 

In  the  development  of  m  in  powers  of  secz  the  coefficient  of 
sec2z  is  independent  of  the  law  of  diminution  of  the  density 
of  the  air  with  its  height. 

It  has  already  been  pointed  out  that  the  Besselian  law  of 
density  of  the  air  gives  a  total  mass  greater  than  the  actual 
mass  as  indicated  by  the  pressure  arid  temperature  at  the  base. 
It  follows  that  in  this  theory  the  coefficient  of  sec2z  is  too  large 
in  the  same  ratio.  To  determine  the  coefficient  in  question  we 
note  that  the  total  mass  of  the  column  of  air  is  the  product  of 
the  density  at  its  base  into  the  pressure-height.  We  therefore 
have 


Making  these  successive  substitutions  we  have 

too 
xdw  =  l, (24) 
o 

and  then  m1  =  j/  — |oc, (25) 

for  all  laws  of  density. 

114.  Passing  to  the  determination  of   the  higher  values  of 
mn,  we  see  from  (22)  and  (23)  that  the  integrals  required  are  all  of 

the  general  form  fi 

I  xnwldw. 
Jo 

To  investigate  the  value  of  this  integral,  we  first  substitute  for 
w  and  dw  their  expressions  in  terms  of  p,  thus  obtaining 


.(26) 
Pi 


214  ASTRONOMICAL  REFRACTION  [§  114. 

Thus  we  have 


Pi  Pi"  -  Pi 

These  terms  are  to  be  integrated  for  w  between  the  limits  1 
and  0.     The  integrals  to  be  found  now  become  of  the  general  form 


........................  (27) 

o 
Interchanging  the  limits  of  integration  as  to  p,  we  then  have 


J^-i/M+-/M-  .............  (28) 

o  l  .  ^ 

The  values  of  7w>Jt  depend  on  the  relation  between  the  density 
and  the  height,  for  which  we  shall  take  the  two  fundamental 
hypotheses  A  and  D. 

115.  Development  on  Newton's  hypothesis. 
Hypothesis  A  gives         ^  —  ^e-  ^ 

dp=  —  ple~xdx', 

fM 
e-lf+ifc&tfx. 
0 

This  is  a  well-known  Eulerian  integral,  of  which  the  value 
may  be  found  by  integrating  by  parts,  leading  to  the  result 


Assigning  to  K  the  successive  values  0,  1,  2  ...  (28)  becomes 

--'+'-  <»> 


This  general  form,  by  assigning  suitable  values  to  n  and  i, 
gives  the  coefficients  of  vnOL\  which  appear  in  the  integration  of 
(22),  and  thus  lead  to  the  following  expressions : 

f. 


I   usdw 

J°  j. (3i) 

Jo 

r. 


=  1 20j/5  - 


§  116,]          DEVELOPMENT  ON   IVOKY'S   HYPOTHESIS  215 

These  being  multiplied  by  the  factors  [i]  give  the  values  of 
m2  .  .  .  m5  on  the  Newtonian  hypothesis. 

116.  Development  on  Ivory's  hypothesis. 

Here  the  relation  between  w  and  x  is  given  by  (16)  and  (17) 
of  §  100.  We  recall  that  the  height  h0  is  that  of  the  absolute 
zero,  supposing  the  temperature  to  go  on  diminishing  at  a 
constant  rate  with  increasing  altitude,  which  it  seems  to  do  up 
to  the  highest  point  to  which  explorations  extend.  We  put 

v-Ao 

"V 

Then  (16)  becomes         £-=(l-?Y    ...........................  (32) 

Pi     v       v/ 

We  now  replace  x  by  another  variable  y, 

_      x 


Then  x  =  v(I-y) 


dx=-vdy 


v~l 


w  =    —  y 
dp  =  (v-l 
We  then  have  for  substitution  in  the  first  member  of  (27), 


(33) 


For  p  =  pl  we  have  y  =  l  and  for  p  =  0  y  =  0.     Substituting 
these  expressions  and  these  limits  of  integration  in  (27),  we  find 


-V+v-2dy  ................  (34) 

o 

This  is  a  Eulerian  integral  which  can  be  evaluated  by  suc- 
cessive integration  by  parts  so  as  to  reduce  the  exponent  n  step 
by  step  to  0.  The  development  of  this  process  belongs  to  the 
integral  calculus.  We  shall,  therefore,  only  state  the  general 
result. 

For  this  purpose  the  F  functions  of  Euler,  or  the  II  functions 
of  Gauss,  come  into  play.  The  only  difference  between  these 
functions  is  one  of  notation, 


216  ASTEONOMICAL  REFRACTION  [§  116. 

The   II   form  of   expression  is  most  convenient  for  our   use, 
because,  when  n  is  a  positive  integer, 


Taking  m  and  n  as  the  exponents,  the  known  general  value 
of  the  integral  is 

(85) 


which  is   easily  computed   when  in  and  n  are  small  positive 
integers. 

Using  this  general  form  in  (27),  we  have 


, 


Although,  in  the  form  in  which  we  have  stated  the  hypothesis, 
v  is  a  function  of  the  temperature,  the  rate  of  diminution  is 
still  so  far  doubtful  that,  practically,  nothing  is  lost  in  the 
present  state  of  our  knowledge  by  using  a  constant  value  of  v. 
We  shall  therefore  put 

v  =  6, 

which  amounts  to  supposing  the  absolute  zero  to  be  reached  at 
6  times  the  pressure  height,  whatever  that  may  be,  and  the  rate 
of  diminution  of  T  with  the  height  to  be  always  5°'6  per 
kilometre. 


•*n,<c  —  **  ' 

For  the  values  0,  1,  and  2  of  K  we  have 

In,Q=  Cr' 
/„  1  =  5-6" 


r     _5.Pn  I*!"! 

n,2  —  &  r 


which  are  the  only  values  we  need  for  our  present  purpose. 


117.]         DEVELOPMENT  ON   IVORY'S  HYPOTHESIS 


217 


By  assigning  to  n  the  special  values  2,  3,  4,  ...  and  substitution 
in  (28),  with  i  =  0,  1,  and  2,  we  find 


x2dw  = 


[X*dw    =  27. 

Jo 


Then,  by  substitution  in  (23), 


.(38) 


£ 

i: 
j: 

Then,  from  (22)  we  have,  including  the  value  of  ml  already 
found, 


On  comparing  the  values  of  the  preceding  integrals  (38)  with 
those  derived  on  the  Newtonian  hypothesis  (31)  it  will  be  seen 
that  the  coefficients  and,  therefore,  the  values  of  m2,  ms,  etc.,  on 
the  latter  hypothesis  are  constantly  larger  than  those  on  Ivory's,, 
which  we  regard  as  the  most  probable.  This  is  quite  in  accord- 
ance with  the  fact,  it  being  found  that  on  the  Newtonian 
hypothesis  the  refraction  is  too  large  as  we  approach  the  horizon, 
at  which  point  it  is  about  one-tenth  greater  than  the  true  value. 

117.   Construction  of  tables  of  refraction. 

The  preceding  completes  the  theory  of  the  development  in 
powers  of  sec  z  so  far  as  the  general  expressions  which  determine 


218  ASTEONOMICAL  REFRACTION  [§  117. 

the  refraction  are  concerned.     It  is,  however,  necessary  to  show 
how  the  results  of  these  expressions  are  put  in  the  special  form 
adopted  in  the  tables  of  refraction  as  described  in  §  98. 

The  practical  method  now  generally  adopted  of  constructing 
such  tables  is  due  to  Bessel.     The  logarithms  of  the  four  principal 
factors  are  tabulated  in  the  following  way  : 

Firstly,  standard  values  of  the  temperature  and  pressure  are 
adopted,  and  for  these  special  values  a  table  giving  the  logarithm 
of  the  refraction  as  a  function  of  the  apparent  zenith  distance 
is  computed.     We  shall  take  as  standards  : 

Temperature,  50°  F.    (Tl.o=281°-5)      | 
Pressure,  30  inches       (Bl  =  762  mm.)  j" 

These  are  near  the  mean  temperature  and  pressure  at  the 
active  observatories,  an  approximation  to  which  is  desirable  in 
choosing  the  standard  temperature. 

By  substituting  the  preceding  values  of  r  and  B  in  (36)  of  §  107, 
we  shall  have  a  standard  density  plt  0  of  the  air.     Putting,  for  the 
time  being,  6r=  6r0,  and  taking  an  arbitrary  standard  temperature 
t^    as  that  of  the  mercury  in  the  barometer  when  it  has  the 
standard  height  ^  =  762  mm.,  the  standard  density  will  become 

_  0-3511  762 

pl'Q~  281-5  '  760(1  +Kt1j  * 

If,  as  is  usual,  we  take  0°  C.  as  the  standard  temperature  for 
the  mercury,  we  shall  have  t±  =  0,  and  the  standard  density  will  be 

^0  =  0-0012505. 

With  this  value  of  Pl  and  c  =  0*226  07  (§  95)  we  find  from  (14), 
a  =  0-00028263  =  58"'297  .................  (4-2) 

The  expression  (16)  for  the  standard  refraction  thus  becomes 

(43) 


where  m0  is  the  value  of  m  for  standard  r  and  B. 

The  general  value  of  m  is  given  by  (21),  where  we  are  to 
substitute  the  values  of  the  coefficients  from  (39).     The  latter 
contain  v  in  (10),  to  compute  which  we  require  the  radius  of 
curvature  a  of  the  geoid.     This  ranges  between 
Ioga  =  6'80l7  at  the  equator 
and  loga  =  6'8061  at  the  poles. 


§  117.]       CONSTRUCTION   OF  TABLES   OF  REFRACTION  219 

For  the  value  of  g,  the  ratio  of  gravity  at  the  place  to  that  at 
Paris,  we  have 

log<7  =  —  0-0013  at  the  equator 

and  \ogg=  +O0010  at  the  poles. 

From  7  =  29-429  m.  (§87)  and  TI  =  281-5  we  now  find 
At  the  equator,  i/  =  0'00l  309, 
At  the  poles,      v  =  O001  292. 

The  difference  between  these  values  is  practically  not  im- 
portant, because  at  low  altitudes,  where  it  might  be  sensible,  the 
refraction  is  necessarily  uncertain.  The  differences  between  the 
curvature  of  the  atmospheric  strata  in  different  latitudes  need 
not  therefore  be  considered  at  present.  We  may  use  for  all 
latitudes  at  standard  r, 

y0= 0-001 30. 

With  this  value  of  v  and  the  corresponding  value  of  oc, 

a0  =  0-000  283, 

we  find  the  numerical  values  of  mlf0,  m2i0,  etc.,  from  (39), 
mlt  o  =  0-001 16, 
m2)  0  =  0-000  001  4, 

etc.,  etc. 

Then  from  (21),  m0=  1  -O'OOl  16 sec20  +  etc. 

At  the  zenith  we  have 

m0  =  0-998  84.* 


*  This  expression  for  the  refraction  diverges  from  that  usually  derived  in 
that  the  latter  is  developed  in  powers  of  tanz  and  the  value  of  m  becomes  1  at 
the  zenith.  The  difference  of  form  arises  from  the  fact  that  the  previous  investi- 
gators have  used  instead  of  the  symbol  h  employed  in  §  104  the  quantity  s,  the 
ratio  of  the  height  h  to  a  +  h,  the  actual  distance  from  the  centre  of  curvature. 
The  value  of  h  thus  appearing  in  the  denominator  complicates  the  theory  and  at  the 
same  time  makes  it  less  rigorous,  because  when  we  neglect  the  higher  powers  of  s 
the  factor  of  the  refraction  depending  on  curvature  vanishes  at  the  zenith.  As  a 
matter  of  fact,  however,  it  does  not  so  vanish,  but  converges  toward  the  finite 
quantity  found  above,  as  can  readily  be  seen  by  geometric  construction.  The 
difference  is,  however,  little  more  than  a  matter  of  form  and  simplicity.  It  is 
easy  to  substitute  the  tangent  for  the  secant  in  the  preceding  developments  ;  but 
nothing  would  be  gained  by  this  course,  except  facilitating  the  comparison  with 
former  theories. 


220  ASTKONOMICAL  KEFRAGTION  [§  117, 

Thus,  for  small  zenith  distances,  we  have,  under  standard 
conditions,  £o  =  58"-230  tan  z. 

This  factor  of  tan  z  is  what  is  properly  called  the  constant  of 
refraction.  We  have  derived  it  by  starting  from  the  observed 
refractive  index  of  air  for  the  brightest  part  of  the  spectrum. 
But  in  practice  it  is  derived  from  observations  of  zenith  distances 
of  the  stars.  The  corresponding  value  of  the  Poulkova  constant  is 

58//-246. 

Reduced  to  gravity  at  the  latitude  of  Paris  this  would  become 

58"-188, 

a  value  slightly  less  than  that  just  computed.  Whatever  the 
adopted  value,  the  table  of  logR  for  standard  conditions  is 
readily  computed. 

118.  The  next  step  will  be  the  tabulation  of  the  logarithms 
of  the  factors  for  the  deviations  of  the  actual  conditions  from 
the  standard  ones.  Returning  once  more  to  §  107  we  see  that, 
at  any  one  station,  pl  contains  three  variable  factors.  Defining 
these  factors  as  those  by  which  we  must  multiply  the  standard 
density  in  order  to  form  the  actual  density,  they  are  : 

1.  Factor  dependent  on  temperature  of  the  external  air, 

281'5 


_ 

TI      271-5  +  Temp.  C" 

2.  Factor  depending  on  barometer, 

}>-**--       B  B 

"^"762  mm.     30  in.'" 

according  to  the  scale  used  on  the  barometer. 

3.  Factor  dependent  on  the  temperature  of  the  mercury, 

(46> 


The  logarithms  of  these  three  factors  are  readily  tabulated. 
They  are  to  be  multiplied  by  factors  depending  on  the  zenith 
distance  and  arising  from  taking  account  of  the  changes  in  the 
values  of  v  and  oc,  and  therefore  in  mv  m2,  etc.,  arising  from  the 


§  118.]       CONSTRUCTION   OF  TABLES  OF  REFRACTION  221 

deviations  from  the  standard  conditions.     To  derive  them  we 
put,  in  (21),  <r  =  ™1sec2s-m2sec40+...,  ..................  (47) 

so  that  m  =  1  —  cr 

and  logm=-3/(cr-icr2+...),  ........................  (48) 

M  being  the  modulus  of  logarithms. 
Putting  er0  for  the  standard  value  of  cr, 

o-0  =  (y0  -  ioc0)  sec20  =  O'OOl  1  6  sec20, 
we  shall  have,  when  we  drop  the  higher  powers  of  cr, 

log  m  —  log  m0  =  M(<rQ  —  or),  ....................  (49) 

from  which  we  may  derive  log  m  when  cr  is  known.  Since  the 
time  of  Bessel  the  universal  practice  has  been  to  develop  <r0  —  cr 
in  powers  of  log  T  and  log  6,  retaining  only  the  first  power. 
This  is  sufficiently  accurate  in  practice  except  near  the  horizon, 
for  which  case  Radau  has  developed  an  improved  method.  To 
show  how  Bessel's  development  is  effected  we  need  only  the 
principal  term  of  cr.  Then  (49)  gives  for  the  reduction  of  log  m 
from  standard  to  actual  conditions 

log  m  —  log  m0  =  M(v0  —  v  +  Joe  —  Joc0)sec%  ..........  (50) 

We  now  have  to  express  v  and  a  in  terms  of  T  and  b. 
Comparing  (8)  and  (10)  with  (44)  and  (45)  we  see  that,  dropping 
insensible  terms, 


T  and  b  being  the  factors  (44)  and  (45).     Thus  we  find 

-=        --)] 
' 


T  and  b  differ  from  1  only  by  a  fraction  of  which  the  average 
value  within  the  range  of  temperatures  at  which  observations 
are  usually  made,  say  —15°  and  +30°,  is  less  than  O'Oo.  To 
quantities  of  the  first  order  as  to  this  difference  we  have 


and  (50)  takes  the  form 

log  m  -  log  m0  ={(z/0  +  Joc0)  log  T+  |oc0  log  b}  sec2z. 


222  ASTEONOMICAL  REFRACTION  [§  118. 

The  corresponding  reductions  of  m2,  ms,  etc.,  may  be  developed 
by  a  similar  process. 

The  use  of  a  refraction  table  will  be  more  convenient  if,  in 
constructing  it,  we  replace  sec%  by  l+tan22  and,  in  the  table 
giving  log  T  and  log  b  as  functions  of  the  temperature  and 
pressure,  multiply  log  T  and  log  b  by  the  constant  factors 


respectively.     Then  we  may  put,  with  Bessel, 


and  tabulate  X  and  A  as  functions  of  z. 

We  now  collect  the  logarithms  of  all  the  factors  which  enter 
into  the  complete  expression  for  the  refraction, 

R  =  am  tan  z, 
as  follows  : 

1.  The  logarithm  of  the  refraction  under  standard  conditions 
or  log  a0m0  tan  z, 

where  a0  =  58"'297(/, 

but  is  subject  to  correction  from  observations,  and 
m0  =  1  —  m1>  0  sec20  +  m2i  0  sec4z  —  .  .  .  , 

the  values  of  the  coefficients  being  taken  from  (39)  with  the 
standard  values  of  v  and  oc. 

2.  The  logarithm  of  the  factor  T,  given  in  (44),  and  tabulated 
as  a  function  of  the  observed  temperature.     This  logarithm  is  to 
be  multiplied  by  the  factor 

X  =  1  +  0-001  44  tan%  +  etc. 

3.  Log  b  in  (45)  multiplied  by  the  factor 

-4  =  1  +  0-000  14  tan2z  +  etc. 

4.  Log  t",  from  (46),  multiplied  by  the  same  factor. 

It  is  to  be  remarked  that  the  values  of  the  factors  X  and  A  are 
here  not  completely  given,  but  only  their  first  terms. 

The  preceding  includes  all  that  is  necessary  to  the  under- 
standing and  intelligent  application  of  the  formulae  and  tables 
of  refraction.  The  completion  of  the  fundamental  theory  with  a 


§  118.]       CONSTRUCTION   OF  TABLES   OF  REFRACTION  223 

view  of  perfecting  the  fundamental  base  of  the  tables  requires 
an  investigation  of  refraction  near  the  horizon,  the  effect  of 
humidity,  and  an  extensive  discussion  of  observations,  none 
of  which  can  be  undertaken  in  the  present  work. 


NOTES  AND  REFERENCES  TO  REFRACTION. 

There  is,  perhaps,  no  branch  of  practical  astronomy  on  which  so  much  has 
been  written  as  on  this  and  which  is  still  in  so  unsatisfactory  a  state.  The 
difficulties  connected  with  it  are  both  theoretical  and  practical.  The 
theoretical  difficulties,  with  which  alone  we  are  concerned  in  the  present 
work,  arise  from  the  uncertainty  and  variability  of  the  law  of  diminution 
of  the  density  of  the  atmosphere  with  height,  and  also  from  the  mathe- 
matical difficulty  of  integrating  the  equations  of  the  refraction  for  altitudes 
near  the  horizon,  after  the  best  law  of  diminution  has  been  adopted.  The 
list  of  modern  writers  on  the  subject  includes  many  of  the  greatest  names 
in  theoretical  and  practical  astronomy,  extending  from  the  time  of  Laplace 
to  the  present.  Among  those  who  have  most  contributed  to  the  advance  of 
the  subject  are, — Bouguer,  Bradley,  Laplace,  Bessel,  Young,  Schmidt,  Ivory, 
Gylden  and  Radau. 

BRUHNS,  Die  Astronomische  Strahlenbrechung,  Leipzig,  1861,  gives  an 
excellent  synopsis  of  writings  on  the  subject  down  to  the  time  of  its 
publication.  Of  these,  the  papers  of  Ivory,  On  the  Astronomical  Refraction, 
Philosophical  Transactions  for  1823  and  1838,  are  still  especially  worthy 
of  study. 

Since  that  time  the  following  Memoirs  are  those  on  which  tables  of 
refraction  have  been  or  may  be  based  : 

GYLDEN,  Untersuchungen  uber  die  Constitution  der  Atmosphare  und  die 
Strahlenbrechung  in  derselben,  St.  Petersburg,  1866-68. 

There  are  two  papers  under  this  title  published  in  the  Memoirs  of  the 
St.  Petersburg  Academy  :  Serie  vii.,  Tome  x.,  No.  1,  and  Tome  xiL,  No.  4. 

They  contain  the  basis  of  the  investigations  on  which  the  Poulkova  tables 
of  refraction  were  based.  They  are  supplemented  by  : 

Beobachtungen  und  Untersuchungen  uber  die  A  stronomische  Strahlenbrechung 
in  der  Nahe  des  Horizontes  von  Y.  Fuss  ;  St.  Petersburg  Memoirs,  Serie  vii., 
Tome  xviii.,  No  3. 

RADAU'S  Memoirs  are  : 

Recherches  sur  la  the'orie  des  Refractions  Astronomiques ;  Annales  de 
1'Observatoire  de  Paris,  Me"  moires,  Tome  xvi.,  1882. 

Essai  sur  les  Refractions  Astronomiques  ;  Ibid.,  Tome  xix.,  1889. 

The  latter  work  is  devoted  especially  to  the  effect  of  aqueous  vapour  in 
the  atmosphere,  and  contains  tables  for  computing  the  refraction. 


224  ASTRONOMICAL  REFRACTION 

Among  the  earliest  refraction  tables  which  may  still  be  regarded  as  of 
importance  are  those  of  Bessel  in  his  Fundamenta  Astronomiae.  They  were 
based  upon  the  observations  of  Bradley.  Bessel  felt  some  doubt  of  the 
constant  of  refraction  adopted  in  these  tables,  which  was  increased  by  un- 
certainty as  to  the  correctness  of  Bradley's  thermometer.  The  results  of  his 
subsequent  researches  are  embodied  in  new  tables  found  in  the  Tabulae 
Regiomontanae,  where  the  constant  of  refraction  of  the  Fundamenta  was 
increased.  These  tables,  enlarged  and  adapted  to  various  barometric  and 
thermometric  scales,  have  formed  the  base  of  most  of  the  tables  used  in 
practical  astronomy  to  the  present  time.  But,  it  has  long  been  known  that 
the  constant  of  refraction  adopted  in  them  requires  a  material  diminution — 
in  fact,  that  the  increase  which  Bessel  made  to  the  constant  of  the 
Fundamenta  was  an  error. 

In  1870  were  published  the  Poulkova  tables,  based  on  the  researches  of 
Gylden  already  quoted,  under  the  title  : 

Tabulae  Refractionum  in  usum  Speculae  Pulcovensis  Congestae,  Petropoli, 
1870. 

These  tables  give  refractions  less  by  '002  85  of  their  whole  amount  than 
those  of  Bessel.  Yet,  the  most  recent  discussions  and  comparisons  indicate 
a  still  greater  diminution  to  the  constant. 

In  this  connection  it  is  to  be  remarked  that  up  to  the  present  time  no 
account  has  been  taken  in  using  tables  of  refraction  of  the  effect  of  the 
differences  between  the  intensities  of  gravity  in  different  latitudes.  Even  if 
the  Poulkova  tables  are  correct  for  the  latitude  of  that  point,  60°,  their 
constant  will  still  need  a  diminution  at  stations  nearer  the  equator. 


CHAPTEE  IX. 


PRECESSION  AND  NUTATION. 

Section  I.    Laws  of  the  Precessional  Motions. 

119.  The  Equinox,  or  the  point  of  intersection  of  the  ecliptic 
and  equator,  may  also  be  defined  as  a  point  90°  from  the  pole  of 
each  of  these  circles.  Hence,  if  we  mark  on  the  celestial  sphere 


FIG.  21. 

P,  the  north  pole  of  rotation  of  the  earth,  or  the  celestial  pole ; 

C,  the  pole  of  the  ecliptic ; 

E,  the  equinox, 

these  points  will  be  the  vertices  of  a  birectangular  spherical 
triangle,  of  which  the  base  PC  is  equal  to  the  obliquity  of  the 
ecliptic. 

Both  the  poles  P  and  C  are  continuously  in  motion.  Hence 
the  equinox  is  also  continuously  in  motion. 

The  motion  of  the  ecliptic,  or  of  the  plane  of  the  earth's  orbit, 
is  due  to  the  action  of  the  planets  on  the  earth  as  a  whole.  It  is 
very  slow,  at  present  less  than  half  a  second  per  year ;  and  its 
direction  and  amount  change  but  little  from  one  century  to  the 

next. 

N.S.A.  p 


226  PEECESSION  AND  NUTATION  [§  119, 

The  motion  of  the  equator,  or  of  the  celestial  pole,  is  due  to 
the  action  of  the  sun  and  moon  upon  the  equatorial  protuberance 
of  the  earth.  The  theory  of  this  action  is  too  extensive  a  subject 
to  be  developed  in  the  present  work,  belonging,  as  it  does,  to  the 
domain  of  theoretical  astronomy.  We  must,  therefore,  limit 
ourselves,  at  present,  to  a  statement  of  the  laws  of  the  motion  as 
they  are  learned  from  a  combination  of  theory  and  observation. 
The  motion  is  expressed  as  the  sum  of  two  components.  One  of 
these  components  consists  in  the  continuous  motion  of  a  point, 
called  the  mean  pole  of  the  equator,  round  the  pole  of  the 
ecliptic  in  a  period  of  about  26  000  years,  which  period  is  not 
an  absolutely  fixed  quantity.  The  other  component  consists  in 
a  motion  called  nutation,  which  carries  the  actual  pole  around 
the  mean  pole  in  a  somewhat  irregular  curve,  approximating  to 
a  circle  with  a  radius  of  9",  in  a  period  equal  to  that  of  the 
revolution  of  the  moon's  node,  or  about  18 '6  years.  This  curve 
has  a  slight  ellipticity,  and  its  irregularities  are  due  to  the 
varying  action  of  the  moon  and  the  sun  in  the  respective  periods 
of  their  revolutions. 

In  the  present  section  we  treat  the  motion  of  the  mean 
pole  P.  This,  and  the  pole  C  of  the  ecliptic,  determine  a  mean 
equinox,  by  the  condition  that  the  latter  is  always  90°  distant 
from  each. 

Precession  is  the  motion  of  the  mean  equinox,  due  to  the 
combined  motion  of  the  two  mean  poles  which  determine  it. 

That  part  of  the  precession  which  is  due  to  the  motion  of  the 
pole  of  the  earth  is  called  luni-solar,  because  produced  by  the 
combined  action  of  the  sun  and  moon.  It  is  commonly  expressed 
as  a  sliding  of  the  equinox  along  some  position  of  the  ecliptic 
considered  as  fixed. 

That  part  which  is  due  to  the  motion  of  the  ecliptic  is  called 
planetary,  because  due  to  the  action  of  the  planets. 

The  combined  effect  of  the  two  motions  is  called  the  general 
precession. 

There  is  no  formula  by  which  the  actual  positions  of  the  two 
poles  can  be  expressed  rigorously  for  any  time.  But  their 
instantaneous  motions,  which  appear  as  derivatives  of  the 


§  121.]  FUNDAMENTAL  CONCEPTIONS  227 

elements  of  position  relative  to  the  time,  may  be  expressed 
numerically  through  a  period  of  several  centuries  before  or  after 
any  epoch.  By  the  numerical  integration  of  these  expressions 
the  actual  positions  may  be  found. 

120.  Fundamental  conceptions. 

In  our  study  of  this  subject  the  two  correlated  concepts  of  a 
great  circle  and  its  pole,  or  of  a  plane  and  the  axis  perpendicular 
to  it, 'come  into  play.  In  consequence  of  this  polar  relation,  each 
quantity  and  motion  which  we  consider  has  two  geometrical 
representations  in  space,  or  on  the  celestial  sphere.  In  treating 
the  subject  we  shall  begin  in  each  case  with  that  concept  which 
is  most  easily  formed  or  developed.  This  is  commonly  the  pole 
of  a  great  circle  rather  than  the  circle  itself.  In  the  case  of  the 
equator  the  primary  concept  is  that  of  the  celestial  pole,  since  it 
is  the  axis  of  rotation  of  the  earth  which  determines  the  equator. 
We  note  especially  in  this  connection  that  CP  is  an  arc  of 
the  solstitial  colure,  and  that  the  equinox  E  is  its  pole.  Either 
of  these  may  be  taken  as  the  determining  concept  for  the 
equinox. 

The  motion  of  the  pole  P  at  any  instant  may  be  conceived 
as  taking  place  on  a  great  circle  G  joining  two  consecutive 
positions  of  P.  The  polar  plane  and  great  circle  of  P  then 
rotate  around  the  axis  and  pole  of  G  as  a  rotation  axis,  and  the 
angular  movement  is  the  same  as  that  of  the  pole  P. 

If  G  remains  fixed  as  the  plane  moves,  the  rotation  axis  of  the 
polar  plane  also  remains  fixed.  But  if  the  pole  moves  on  a 
curve  other  than  a  great  circle,  the  rotation  axis  moves  also, 
rotating  around  the  instantaneous  position  of  the  moving  pole 
as  a  centre. 

121.  Motion  of  the  celestial  pole. 

C  and  P  being  the  respective  poles  of  the  ecliptic  and  equator, 
the  law  of  motion  of  the  pole  of  the  equator,  as  derived  from 
mechanical  theory,  is : 

The  mean  pole  moves  continually  toward  the  mean  equinox 


228 


PEECESSION   AND  NUTATION 


[§  121. 


of  ike,  moment,  and  therefore  at  right  angles  to  the  colure  CP, 
with  speed  n  given  by  an  expression  of  the  form 

n  =  P  sin  e  cos  e, (1) 

p  being  a  function  of  the  mechanical  ellipticity  of  the  earth, 
and  of  the  elements  of  the  orbits  of  the  sun  and  moon,  and  e  the 
obliquity  of  the  ecliptic. 

P  is  subject  to  a  minute  change,  arising  from  the  diminution 
of  the  eccentricity  of  the  earth's  orbit;  but  the  change  is  so 
slight  that,  for  several  centuries  to  come,  it  may  be  regarded  as 
an  absolute  constant.  The  writer  has  called  it  the  precessional 
constant*  Taking  the  solar  year  as  the  unit  of  time,  its 
adopted  value  is 

p  =  54"-9066 (2) 

Its  rate  of  change  is  only  —  0"'000  036  4  per  century. 

The  centre  G  of  the  motion  thus  defined  is  the  instantaneous 
position  of  the  pole  of  the  ecliptic  at  the  moment.  This  pole  is 
continually  in  motion  in  the  direction  CC',  as  shown  by  the 
dotted  line  in  Figure  22.  Hence,  at  the  present  time,  the  pole  of 


FIG.  22. 

the  ecliptic  is  approaching  that  of  the  equator.  It  follows  from 
the  law  as  defined  that  if  the  pole  of  the  ecliptic  were  fixed  in 
position,  the  obliquity  would  be  constant.  But,  as  the  pole 
moves,  it  does  not  carry  the  pole  of  the  earth  with  it,  the  motion 
of  the  latter  being  determined  by  the  instantaneous  position 
of  the  pole  C,  unaffected  by  its  motion.  Because  the  pole  C  is 


*This  term  has  been  also  applied  to  what  may  be  called  the  mechanical 
ellipticity  of  the  earth,  or  the  ratio  of  excess  of  its  polar  over  its  equatorial 
moment  of  inertia  to  the  polar  moment. 


§  122.]  MOTION  OF  THE  CELESTIAL  POLE  229 

at  present  diminishing  its  distance  from  P,  the  obliquity  of  the 
ecliptic  is  also  diminishing. 

The  speed  n  of  the  motion  of  the  pole  P,  as  we  have  expressed 
it,  is  measured  on  a  great  circle.  To  find  the  angular  rate  of 
motion  round  0  as  a  centre,  we  must  divide  it  by  sine,  which 
will  give  the  speed  of  luni-solar  precession,  We  therefore  have, 
still  taking  the  year  as  the  unit  of  time : 

Annual  motion  of  P,  actual ;          n  =  54"-9066  sin  e  cos  e\       ,o\ 
Resulting  luni-solar  precession ;    p  =  54"'9066  cos  e         / 

Neither  n  nor  p  is  an  absolute  constant,  since  they  both  change 
with  e,  the  obliquity  of  the  ecliptic. 

122.  Motion  of  the  ecliptic. 

Although  the  position  of  the  ecliptic  is  to  be  referred  to  the 
equator  and  the  equinox,  so  that  the  motion  of  the  latter  enters 
into  the  expression  for  that  position,  yet  the  actual  motion  of 
the  ecliptic  is  independent  of  that  of  the  equator.  We,  there- 
fore, begin  by  developing  the  position  and  motion  of  the  ecliptic, 
taking  its  position  at  some  fixed  epoch  as  a  fundamental  plane. 
Any  such  position  of  its  plane  is  called  the  fixed  ecliptic  of  the 
date  at  which  it  has  that  position. 

The  curve  0(7  along  which  the  pole  of  the  ecliptic  is  moving 
in  our  time  is  not  a  great  circle,  but  a  curve  slightly  convex 
toward  the  colure  OP.  To  make  clear  the  nature  and  effect  of 
this  motion  we  add  Fig.  23,  showing  the  correlated  motion  of 
the  ecliptic  itself.  This  represents  a  view  of  the  ecliptic  seen 
from  the  direction  of  its  north  polar  axis.  The  positions  of  the 
poles  P  and  G  are  reversed  in  appearance,  because  in  Fig.  22 
they  are  seen  as  from  within  the  sphere,  while  in  Fig.  23  they 
are  seen  as  from  without. 

We  shall  now  explain  the  motion  by  each  of  these  correlated 
concepts.  As  the  pole  G  moves,  the  ecliptic  rotates  around  an 
axis  NM  (Fig.  23)  in  its  own  plane,  determined  by  the  condition 
that  N  is  a  pole  of  the  great  circle  joining  two  consecutive 
positions  of  the  pole  G.  From  the  direction  of  the  motion  it 
will  be  seen  that  the  axis  N,  which  we  have  taken  as  funda- 
mental, is  at  each  moment  the  descending  node  of  the  ecliptic, 


230 


PKECESSION  AND  NUTATION 


[§  122. 


while  M  is  the  ascending  node.  The  curve  (7(7  being  convex 
toward  CP,  the  node  N  is  slowly  moving  in  the  retrograde 
direction  from  E  toward  L. 


FIG.  23. 


Fig.  24  shows  the  effect  of  this  motion  of  the  ecliptic  upon 
the  position  of  the  equinox,  supposing  the  equator  to  remain 
fixed.  Here  EN  is  the  ecliptic  and  EQ  the  equator,  as  seen  from 
the  centre  of  the  sphere,  the  observer  at  C  in  Fig.  23  looking  in 
the  direction  E.  The  rotation  of  the  ecliptic  around  N  is  con- 


FIG.  24. 

tinuous  from  NE  toward  NEV  causing  the  equinox  on  the  fixed 
equator  to  move  in  the  positive  direction  EE^  thus  increasing 
the  angle  EN.  This  motion  is  that  of  planetary  precession. 

In  consequence  of  luni-solar  precession  the  colure  CP  is 
rotating  around  the  instantaneous  position  of  C  as  an  axis, 
carrying  with  it  its  pole,  the  equinox  Et  in  the  direction  EL 
with  a  motion  yet  more  rapid  than  that  of  N.  The  angle  ECN 
is  therefore  diminishing. 


§  123.]  MOTION  OF  THE  ECLIPTIC.  231 

The  instantaneous  motion  of  the  ecliptic  is  defined  by  the 
speed  of  its  rotation  around  the  axis  MN,  which  speed  we  call 
K,  and  by  the  position  of  N  relative  to  the  equinox.  We  put : 

NO,  the  angle  between  the  direction  of  motion  CC',  as  seen  in 
Fig.  22,  and  some  fixed  position  of  the  colure,  say  that  of  1850, 
which  we  call  the  initial  colure  and  date.  The  correlated 
concept  is  the  arc  E^N  (Figs.  23  and  24). 

N,  the  angle  between  this  direction  at  the  epoch  t  and  the 
colure  at  t.  This  is  equivalent  to  saying  that  NQ  is  the  angle  which 
the  tangent  to  the  curve  CC'  makes  with  the  colure  of  the  initial 
date,  while  N  is  the  angle  which  it  makes  with  the  actual  moving 
colure.  The  correlated  N  is  the  arc  EN  (Figs.  23  and  24). 

These  quantities  determine  only  the  instantaneous  motion, 
not  the  actual  position  of  the  ecliptic.  To  express  the  latter 
we  shall  hereafter  put 

k,  the  angle  CC'  (Fig.  W)  =  ENE^  (Fig.  24),  which  the  actual 
ecliptic  at  any  epoch  makes  with  the  initial  ecliptic  or  funda- 
mental plane. 

JVp  the  angle  which  the  node  of  the  actual  ecliptic  makes  with 
the  initial  line  of  the  equinoxes. 

In  the  usual  method  of  expressing  the  position  of  the  moving 
with  respect  to  a  fixed  ecliptic,  k  is  the  inclination,  and  180°—^ 
the  longitude  of  the  ascending  node,  referred  to  the  initial 
equinox.  The  value  of  N  at  present  being  6°  and  a  fraction, 
the  longitude  of  this  node  is  173°  and  a  fraction. 

123.  Numerical  computation  of  the  motion  of  the  ecliptic. 

Proceeding  to  the  numerical  computation,  the  speed  of  the 
instantaneous  motion  and  the  values  of  N0  are  found  by  theory 
to  be  as  follows  at  three  epochs,  of  which  the  extremes  are  250 
years  before  and  after  1850.* 

Epoch.  log/c.  K.  N0.  KcosNp 

1600  1-67500  47"-316  5°  17'-96  47-113 

1850  1-67340  47-141  630-32  46-838         5-341  }-.  (4) 

2100  1-67187  46-976  742-82  46-550 


*  Astronomical  Papers  of  the  American  Ephemeris,   vol.    iv. ;  Elements  and 
Constants,  p.  186. 


232  PRECESSION   AND  NUTATION  [§  123. 

Our  next  step  is  to  derive  from  (4)  the  actual  position  of 
the  ecliptic,  at  any  intermediate  epoch.  This  we  do  by  referring 
the  position  of  the  pole  G  to  rectangular  coordinates,  the 
curvature  of  the  sphere  within  so  minute  a  region  as  that  over 
which  the  motion  extends  being  insensible.  Taking  GP  as  the 
axis  of  F  and  x,  y  as  the  coordinates  of  C,  we  shall  have 

K  sin  Nn 


Putting  T  for  the  time  in  centuries  after  1850,  the  three 
values  of  these  quantities  already  given  may  be  developed  in 
the  form  : 

D^  =   5"-341  +  0"-38702T-  (TOGO  56T2, 
DTy  =  46"-838  -  0"'1I26T-  (TOOl  04T2. 

Then,  by  integration, 

x  =   5'S34ir+0''-1935T2--0"-00019r3l  (6> 

y  =  46"'838T-  (T-0563272  -  (T'OOO  35T3J  ' 

Here  x  and  y  are  the  coordinates  of  the  pole  G  referred  to 
the  colure  of  1850  as  a  fixed  direction.  To  find  the  polar 
coordinates,  we  put 

C',  the  position  of  the  pole  at  any  epoch  ; 
k,  the  arc  of  the  great  circle  GG'  ; 
Nv  the  angle  PCC'. 

The  values  of  k  and  ^  at  any  time  are  then  found  from  the 
equations  ksmN^x, 

k  cos  N!  =  y. 

Computing  the  values  of  x  and  y  from  (6)  for  epochs  fifty 
years  apart,  we  have  the  results  shown  in  the  following  table  : 


Epoch. 
1750 

X. 

-  5"-147 

y- 
-  46"-894 

k. 
-  47"-176 

6°  15'-81^ 

1800 

-  2  -622 

-  23  -433 

-  23  -579 

6  23-07 

1850 

0  -000 

0  -000 

0  -000 

6  30-32 

1900 

+  2  -719 

+  23  -405 

23  -562 

6  37-55 

1950 

5  -534 

46  -781 

47  -107 

6  44-79 

2000 

8  -446 

70  -129 

70  -636 

6  52-04 

2050 

11  -454 

93  -448 

94  -147 

6  59-28 

2100 

14  -558 

116  -738 

117  -642 

7  6-52' 

/7\ 


§  124.]        COMBINATION  OF  PRECESSION AL  MOTIONS  233 

In  this  table  the  value  of  N:  for  the  initial  epoch  1850  is  the 
direction  of  the  instantaneous  motion  at  that  epoch.  For  con- 
venience in  subsequent  computation  the  value  of  k  is  regarded 
as  negative  before  1850,  thus  avoiding  a  change  of  180°  in  Nr 

124.  Combination  of  the  precessional  motions. 

We  have  now  to  combine  the  two  motions  which  we  have 
defined,  so  as  to  obtain  the  general  precession.  We  begin,  as 
before,  with  the  speeds  of  the  motions  and  not  with  their  total 
amount  between  two  epochs.  This  speed  is  given  by  the  motion 
during  a  time  so  short  that  we  may  regard  the  motion  as 
infinitesimal,  but  may  be  expressed  with  reference  to  any  unit 
of  time  that  we  find  convenient. 

If  we  define  the  motion  by  that  of  the  two  poles,  the  annual 
general  precession  is  equal  to  the  annual  change  in  the  direction 
of  the  colure  PC,  as  measured  by  the  rotation  around  the 
point  C.  But  the  effect  of  the  combined  motions  on  the  position 
of  the  actual  equinox  can  best  be  studied  by  transferring  our 
field  of  view  from  the  region  of  the  poles  to  that  of  the  equinox, 
and  studying  the  motion  of  the  ecliptic  and  equator  themselves, 
instead  of  the  motion  of  their  poles. 


I — Q° 


FIG.  25. 

Fig.  25  is>a  view  of  the  moving  equinox,  seen  from  the  same 
view-point  as  in  Fig.  24,  but  infinitely  magnified. 
In  Fig  25,  let  us  have : 

QR,  the  position  of  the  equator ; 

LM,  that  of  the  ecliptic ; 

E,  the  equinox. 

Two  positions  of  each  of  these  are  marked,  the  one  set  Q0^o  '•>• 
LQMQ ;  EQ ;  for  the  origin  or  zero  of  time  ;  the  other,  QR ;  LM ;  E ;, 


234  PEECESSION  AND  NUTATION  [§  124. 

after  a  period  of  time  which  we  regard  as  infinitesimal.  All  the 
segments  in  the  figure  are,  therefore,  treated  as  infinitesimals, 
and  are  considered  to  represent  speeds  of  motion,  each  speed 
being  multiplied  by  dt. 

We  now  apply  what  has  already  been  said  of  the  motion  of 
the  poles  to  the  figure,  with  the  following  results  : 

The  two  equators  Q0R0  and  QR  intersect  at  points  90°  in 
•either  direction  from  the  region  shown  in  the  figure,  and  their 
infinitesimal  arcs  shown  in  the  figure  are  parallel. 

The  perpendicular  distance  ES  of  the  two  equators  from  each 
other  is  equal  to  ndt  ;  but,  in  accordance  with  what  has  just 
been  said,  we  may  consider  this  distance  to  represent  n  itself, 
the  factor  dt  being  dropped. 

The  two  ecliptics  LoM0  and  LM  intersect  at  the  point  N,  which 
cannot  be  marked  in  the  figure,  lying  in  the  direction  LM  at  a 
distance  from  E0  (or  E)  represented  by  the  angle  N  already 
defined. 

The  speed  p  of  the  luni-solar  precession  is  represented  by 
the  arc  EQEl  between  the  intersections  of  the  two  equators  with 
the  fixed  ecliptic  L0M0. 

The  arc  FE0  may  be  called  the  luni-solar  precession  in  R.A. 
Its  value  is  p  cos  e  or  P  cos2e,  but  it  is  not  used  by  itself. 

The  arc  EEl  on  the  equator  is  the  planetary  precession  in  R.A. 
We  call  it  A'.  The  speed  of  rotation  of  the  ecliptic  around  the 
node  N  being  /c,  we  have 


The  total  speed  of  precession  in  R.A.  is 

E0S  =  E0F-  EE,  =  P  cos2e  -  A'. 

The  general  precession  is  defined  as  the  motion  of  the  equinox 
E  along  the  moving  ecliptic.  It  is  measured  by  its  projection 
EQT,  which  differs  from  E0E  only  by  an  infinitesimal  of  the 
second  order.  Its  two  parts  are  p  =  E0El  and  E1T=\'cose 
taken  negatively.  We  call  its  speed  I.  Hence 

l=p  —  \'cose  =  (P  —  A')  cose  ...................  (8) 

From  the  law  of  motion  of  the  equator,  P  always  moving  at 
right  angles  to  CP,  it  will  be  seen  that  the  instantaneous  change 


§  125.]        COMBINATION   OF  PEECESSIONAL  MOTIONS  235 

of  the  obliquity  is  due  wholly  to  the  motion  of  the  ecliptic,  and 
may  be  found  by  resolving  the  instantaneous  motion  of  C  into 
two  rectangular  components,  one  in  the  direction  CP ;  the  other 
in  the  direction  GE.     (Figs.  22,  23.) 
Since,  by  the  preceding  notation, 
N=  angle  PCC', 
K  —  rate  of  motion  of  (7, 
we  shall  have  Dte  =  —K cos N. 

125.  Expressions  for  the  instantaneous  rates  of  motion. 

As  the  conceptions  developed  in  the  preceding  sections  are 
fundamental  in  spherical  astronomy,  we  recapitulate  them. 
Dropping  the  factor  dt  and  supposing  the  lines  in  the  figure 
to  represent  rates  of  motion,  the  perpendicular  distance  SJE 
or  FEl  at  the  equinox  between  the  two  positions  of  the  equator 
will  represent  n.  The  distance  ET  between  the  ecliptics  will 
represent  K  sin  N.  We  then  have 

speed  of  luni-solar  precession  in  longitude, 

p  =  EQEl  =  Pcose',  (9) 

speed  of  planetary  precession  in  longitude, 

-\'cose  =  ElT=  -K  sin  N  cote; (10) 

speed  of  general  precession  in  longitude, 

Z=2>-X'cose  =  (P-\')co8e;  (11) 

speed  of  luni-solar  precession  in  R.A., 

E0F=N1E=pcose  =  Pcotfe; (12) 

speed  of  planetary  precession  in  R.A., 

-\'  =  EE1=  -/csin  JVcosece; (13) 

speed  of  general  precession  in  R.A., 

m  =  p  cos2e  —  X' ;  (14) 

speed  of  change  of  the  obliquity  of  the  ecliptic, 

(15) 


236  PRECESSION  AND   NUTATION  [§  126. 

126.  Numerical  values  of  the  processional  motions  and  of  the 
obliquity. 

We  shall  now  compute  from  the  data  already  given,  and  the 
preceding  formulae,  the  actual  speeds  of  the  various  precessional 
motions  for  some  fundamental  epochs.  We  have  all  the  data 
for  1850  at  hand ;  but,  for  the  other  epochs,  it  is  necessary  to 
use  the  results  for  1850  to  compute  the  data.  These  are  the 
values  of  P  and  N  already  given,  and  the  obliquity,  of  which 
the  value  for  1850  is 

e  =  23°  27'31"'ti8. 

For  this  class  of  computations  the  century  is  the  most  con- 
venient unit  of  time;  we  therefore  multiply  the  value  of  P  by 
100,  so  that  P  =  5490"-66. 

The  computation  is  as  follows  : 

EPOCH  1850. 

log  P     3-739  624  5 

cose     9-9625334. 

log  p     3-7021579 

sine     9-5999808 

„     ^cose     3-6646913 

n     3-3021387 

Luni-solar  precession  p  =  5036" -84 

Luni-solar  precession  in  K.  A.  =  4620  "53 

^  =  2005  -11 
=    133"674 


log    cosN0     9-99719 
K     1-67340 
„      sin  N0     9-054  21 
„     cosece    0-400  02 
log          A'     1-12763 
„     A'  cos  e     1-09016 
A'  =  13"-416 
A' cose  =12  -307 

General  precession  in  long.     I  =  5024"-53  =   83'  -742 
General  precession  in  E. A.    m  =  4607  -11  =  307"141 
log  K  cos  N0  =1-670  59 
De  =  -46"-837 


§  126.]    NUMERICAL  VALUES  OF  PRECESSIONAL  MOTIONS  237 

We  have  next  to  derive  the  data  and  compute  the  speeds  of 
motion  for  the  extreme  fundamental  epochs.  N0  being  the 
angular  distance  of  the  instantaneous  axis  of  rotation  from  the 
equinox  of  1850,  and  N  that  from  the  actual  equinox,  it  follows 
that  their  speeds  differ  by  the  general  precession  in  longitude, 
so  that  we  have 

DtN=DtN0-l 

By  developing  the  values  (4)  of  N0,  we  have 


We  have  just  found         I  =  83'742. 
Therefore,  postponing  terms  in  T2,  we  have 


from  which  we  derive  N  for  other  epochs. 

With  these  expressions,  and  the  values  of  K  derived  from  (4) 
by  interpolation,  we  compute  the  following  values  of  the  quan- 
tities required  to  find  the  obliquity  of  the  ecliptic  and  the 
planetary  precession : 

Epoch.  N.  log*.  KCosN.  KsinJV. 

1750         7°25'-09         1-67403         46"-814 


(16) 


1800 

6 

57 

•71 

1-673 

71 

46 

•828 

5 

•718 

1850 

6 

30 

•32 

1 

•673 

40 

46 

•837 

5 

•341 

1900 

6 

2 

•94 

1 

•67309 

46 

•844 

4 

•964 

1950 

5 

35 

•55 

1 

•672 

78 

46 

•849 

4 

•587 

2000 

5 

8 

•17 

1 

•672 

47 

46 

•851 

4 

•211 

2050 

4 

40 

•78 

1 

•672 

17 

46 

•851 

3 

•835 

2100 

4 

13 

•40 

1 

•671 

87 

46 

•848 

3 

•459- 

Differencing  the  preceding  K  cos  N,  the  centennial  variation  of 
the  obliquity,  we  find  that  its  second  differences  are  appreciably 
constant,  and  that  its  values  may  be  developed  in  the  form, 

K  cos  N  =  -  DTe  =  46//<837  +  0"-01 7T-  0"'0051T2. 

By  integrating  and  adding  as  a  constant  the  obliquity  for 
1850,  we  find 

e=23°  27'  31"'68-46'/-837T-0/'-0085T2+0''-00177T3. 


238  PBECESSION  AND  NUTATION  [§  126. 

This  gives  the  following  values  of  the  obliquity  for  the  eight 
epochs  from  1750  to  2100 : 


Epoch. 
1750 

€. 

23°  28'  18"-51 

log  sin  e. 

9-600  207  9 

log  cos  e. 
9-962  490  7x 

1800 

23    27  55  -10 

9-6000943 

9-9625121 

1850 

23    27  31  -68 

9-5999808 

9-9625334 

1900 

23    27     8  -26 

9-5998671 

9-9625548 

(17) 

1950 

23    26  44  -84 

9-5997534 

9-9625762 

'••\**/ 

2000 

23    26  21  -41 

9-5996396 

9-9625976 

2050 
9100 

23   25  57  -99 

9,3    9fi    5U  -fifi 

9-5995259 

Q-RQQ  11  9  1 

9-9626190 

Q'Q«9  filH  KJ 

These  quantities  give  the  data  necessary  for  the  computation 
of  all  the  precessional  motions  for  the  several  epochs.  Another 
approximation  to  the  values  for  epochs  before  and  after  1850 
may  be  made  by  using  the  varying  values  of  I  to  derive  fresh 
values  of  N.  But  this  revision  will  not  appreciably  change  the 
results  which,  so  far  as  necessary  for  use  during  the  present 
century,  will  be  found  in  Appendix  III. 


Section  II.    Relative  Positions  of  the  Equator  and  Equinox 
at  Widely  Separated  Epochs. 

127.  Our  next  problem  is,  from  the  instantaneous  motions 
just  found,  to  define  the  actual  position  of  the  equator  at  any 
one  epoch  T  relative  to  its  position  at  some  other  epoch  T0.  We 
regard  T0  as  a  constant,  and  term  it  the  initial  epoch.  The 
other,  with  the  quantities  which  depend  upon  it,  are  treated  as 
variable. 

Let  us  consider  the  spherical  quadrangle  PP0G0C  formed  by 
the  positions  of  the  two  poles  at  the  two  epochs. 

In  order  to  represent  all  the  quantities  on  the  figure,  it  has 
been  necessary  to  draw  it  so  as  to  express  the  motion  over  a 
period  of  several  thousand  years.  In  consequence  of  this  the 
angle  CC0P,  as  represented  in  the  figure,  is  negative,  owing  to 
the  motion  of  the  pole  P  having  carried  the  arc  C0P  over  the 
point  C.  The  student  who  wishes  to  do  so  can  easily  draw  the 


§  127.]    RELATIVE  POSITIONS  OF  EQUATOR  AND  EQUINOX   239 

figure  and  apply  the  numbers  for  the  period  through  which  the 
computations  actually  extend. 

GO 


FIG  26. 

We  divide  the  quadrangle  into  two  triangles  by  the  diagonal 
C0P,  and  then  have  or  put 

e0  =  CoP0,  the  obliquity  of  the  ecliptic  at  the  initial  epoch. 

€l  =  C^P,  the  obliquity  of  the  equator  of  the  epoch  T  to  the 

initial  ecliptic. 
0,  the  arc  P0P  joining  the  two  positions  of  the  pole.     This  arc 

is  to  be  taken  as  that  of  a  great  circle,  not  the  actual 

path  of  P,  which  is  represented  by  the  dotted  arc. 
k,  the  arc  C0C,  through  which  the  pole  of   the  ecliptic  has 

moved. 

Nlt  the  angle  P0C00. 

f,  the  amount  by  which  the  angle  00PP0  falls  short  of  90°. 
fo,  the  amount  by  which  the  angle  (70P0P  falls  short  of  90°. 
A,  the  angle  C^PC,  which  is  equal  to  the  total  planetary 

precession  on  the  equator  of  the  epoch  T,  or  to  the  arc 

of  this  equator  intercepted  between  the  two  ecliptics, 

taken  negatively  in  the  figure.* 

*It  should  be  noted  that  the  angle  X,  when  taken  positively,  as  is  done  in  the 
present  work  for  dates  subsequent  to  the  initial  epoch,  is  subtract! ve  from  the 
lunar  solar  precession  during  the  next  500  years.  Its  value  will  reach  a  maximum 


240  PRECESSION   AND  NUTATION  [§127. 

0,  the  amount  by  which  the  angle  PQPC  falls  short  of  90°, 
so  that  we  have  .-• 

*-/-x 

t/r,  the  angle  P00(>P,  the  total  luni-solar  precession  on  the  fixed 

position  of  the  initial  ecliptic. 
T,  the  interval  after  the  initial  epoch,  in  terms  of  100  solar 

years  as  the  unit  of  time. 

To   find  the  derivatives,  or   instantaneous   motions  of  these 
various  quantities,  we  have  to  suppose  two  consecutive  positions 
of  the  poles  P  and  C,  the  second  of  which  we  call  P'  and  (7, 
and  apply  the  differential  formulae  of  the  last  section. 
Since,  by  definition, 

Angle  PoPC=  90°  -0, 

while,  by  the  fundamental  law  of  the  motion,  P  moves  at  right 
angles  to  CP,  it  follows  that  PP'  makes  the  angle  z  with  the 
direction  P0P  continued,  and  the  angle  90°  +  X  with  the  arc  PC. 
Its  rate  of  motion  being  n,  the  space  over  which  it  moves  in  an 
infinitesimal  time  is  ndt.  Hence 

PP'  =  mft  =  P  sine  cos  e<&  (§121). 
We  then  have,  as  the  effect  of  the  motion  on  0  and  €V 

dO=P0P/-P0P  =  ncoszdt'\  Q8, 

d€l  =  Cf0P/  -C0P  =  n  sin  \dt  j    ' 

The  instantaneous  motion  of  i/r  is  the  angle  subtended  by  PP', 
or  the  motion  ndt,  as  seen  from  (70.     We  therefore  have,  by  the 
theorems  of  differential  spherical  trigonometry  (§  7), 
ndt  cos  X  =  sin  C0Pd\/s  =  sin 

TT  *    .        n  COS  X 

Hence  Dt\Lr  = 


or 


Between  the  parts  of  the  triangle  G0PC  we  have  the  relation 
sinP  _  sin  C0 
sinGf0(7~smGF 
sinX     sinj  —  ir 


•  .    ,      -  ; 
sin  k  sm  e 


about  2150  and  then  diminish  until  about  2400,  when  it  will  become  negative 
through  the  arc  (70P  crossing  (70(7.  It  should  also  be  seen  that  the  precessional 
motion  X'  defined  in  the  preceding  section,  is  the  speed  of  X  at  the  initial  epoch, 
but  is  not  rigorously  equal  to  that  speed  at  other  epochs. 


§  128.]    EEL  ATI  VE  POSITIONS  OF  EQUATOR  AND  EQUINOX   241 

Owing  to  the  minuteness  of  X  and  k,  we  may  take  their  sines 
.and  arcs  as  equal,  and  put  cosX=l.  Substituting  for  n  its 
value,  we  have  the  following  three  equations  for  X,  elt  and  i/r : 

k  sin  (Nj_  —  \fs) 

X  — : 

sine 
J)t€l  =  p  sin  e  cos  e  sin  X 

P  sin  e  cos  e     P  sin  2e 


.(19) 


sin  ex  2  sin  ex 

The  only  unknown  quantities  in  these  equations  are  X,  el5  and 
T/r,  the  values  of  which  we  are  now  to  derive  by  successive 
approximations. 

128.  Numerical  approximations  to  the  position  of  the  pole. 

At  first  we  need  only  a  rough  value  of  X  to  be  used  in 
finding  e^  For  this  purpose  it  will  suffice  to  suppose  ^  to  vary 
uniformly  with  its  motion  for  1850,  which  is  P  cos  e0.  We  may 
therefore  put,  in  the  first  equation, 


With  the  values  of  \}s  computed  from  this  expression,  and  the 
values  found  in  §§  123  and  126  for  k,  Nv  and  e,  we  compute 
approximate  values  of  X  as  follows : 

1850  ^  =  0°    0/'°  A=   °"'00 

1900  0   42-0  6  -11 

1950  1    23-9  11  -03 

2000  25-9  14  -76 

2050  2    47-8  17  -30 

2100  3    29-8  18  '65 

Changing  the  values  of  X  to  arc,  and  taking  the  6th  decimal 
as  a  unit,  we  find  that  sin  X  may  be  expressed  in  the  form  : 

106  sin  X  =  65-03T- 1 1'54T2. 
Substituting  this  expression  in  (19),  we  have 

Dfr  =  0"-1304r-  0"-0232T2, 
and  by  integration, 

e1  =  eo+0"-0652r2-0"-0077T3 (20) 

With  the  values  of  ex  derived   from   this  expression,  which 

differ  by  only  a  small  fraction  of  a  second  from  e0,  we  find  from 
N.S.A.  Q 


242 


PRECESSION   AND   NUTATION 


[§  128. 


the  third  equation  (19),  using  the  values  of  e  in  (17),  the 
following  values  of  the  differential  variation  of  \^  with  its. 
differences : 


Epoch. 

ay 
dT' 

*v 

Ao.      \ 

1750 

5038"-97 

-  1"-06 

1800 

5037  -91 

-1  -07 

-•01 

1850 

5036  -84 

-1  -06 

+  •01 

i  ar\f\ 

prrvor     .'TO 

•AT 

iyuu 

OUoO     10 

-1  -07 

—  01 

1950 

5034  -71 

-1  -08 

-•01 

2000 

5033  -63 

-1  -08 

•00 

2050 

5032  -55 

-1  -09 

-•01 

(21) 


2100         5031  -46 
These  values  may  be  developed  in  the  form 

^  =  5036"'S4  -  2"-130r-  0"'010T2. 
at 

Hence,  by  integration, 

^  =  5036//-84T-  1"-065T2  -  C 


This  is  the  definitive  expression  for  the  total  precession  of 
the  equinox  upon  the  fixed  ecliptic  of  1850. 

129.  Numerical  value  of  the  planetary  precession. 

We  now  have  all  the  data  for  computing  the  definitive  values 
of  X  from  the  first  equation  (19),  as  follows : 


1"-188 
1  -190 
1  -192 
1  -191 


Epoch. 

y.      NI  -  \f/.      x. 

A, 

1750 

-1° 

23' 

•96 

7° 

39' 

•77 

-15 

"•794 

+  8"- 

491 

1800 

-0 

41 

•98 

7 

5 

•04 

—  7 

•303 

7  • 

303 

1850 

0 

•00 

6 

30 

•32 

0 

•000 

6  • 

113 

1900 

+  0 

41 

•97 

5 

55 

•58 

+  6 

•113 

4  • 

921 

1950 

1 

23 

•93 

5 

20 

•86 

11 

•034 

3  • 

730 

2000 

2 

5 

•88 

4 

46 

•16 

14 

•764 

2  • 

537 

2050 

2 

47 

•82 

4 

11 

•46 

17 

•301 

1  • 

343 

2100 

3 

29 

•76 

3 

36 

•75 

18 

•644 

-1  -194 


§  130.]  NUMERICAL  VALUE  OF  PLANETARY  PRECESSION     243 

The  values  of  X  may  be  developed  in  the  form  : 

X  =  13//-416r-2//-380Ti!-0//-0014!r8  .............  (23) 

This  expression  represents  all  the  above  special  values  within 
(TOOL 

130.  Auxiliary  angles. 

For  the  angles   f0  and   f,  we   have,  from  Napier's  analogies, 
applied  to  the  triangle  P^G^P  : 


The  arc  e1  —  e0  is  only  a  small  fraction  of  1".     We  may  there- 
fore neglect  its  powers.     Putting 

Ae1  =  e1-e0, 

the  above  equations  may  be  written 

^i 


tan  J(f-  £>)  = 


^tan  J^J 


The  computation  of  the  first  of  these  expressions  involves 
no  difficulty,  all  the  quantities  which  enter  into  it  having  been 
found.  The  results  are  as  follows  : 


Epoch. 

|^r. 

Mf*jf 

o)-                        Aj. 

A2.               A3. 

1750 

-  2518"-951 

-  2310" 

•769        1155".519 

1800 

-1259  -343 

-1155 

•257        1155  <257 

-255        +    9 

1850 

0  -000 

0 

•°00        1155  -Oil 

-246            ^ 

1900 

+  1259  -076 

1155 

-Oil 

6 

1950 

2517  -886 

2309 

•793        n54       " 

17 

2000 

3776  -426 

3464 

•352                     5 

1  1  fJ^x      OUO 

18 

2050 

5034  -698 

4618 

•705        11K/(    ir. 
1154  *16o 

-188 

2100 

6292  -698 

5772 

•870 

Developing  these  quantities  in  powers  of  T,  we  find 

f  +  f0  =  4620/r'53r-  0"-984T2  +  O^OSGT3  ..........  (25) 

The  second  of  the  equations  (24)  is  subject  to  the  inconvenience 
of  giving  f—  £0  as  the  quotient  of  two  small  quantities.     This, 


244  PRECESSION   AND  NUTATION  [§  130. 

makes  it  necessary  to  express  the  numerator  of  the  fraction  in 
a  form  to  facilitate  its  development.  By  eliminating  sin  A  from 
the  first  two  equations  ( 1 9),  we  find 

Dtel=Pkcosesiu(N1-\ls) (26) 

In  using  this  formula  we  express  P  in  arc,  and   thus   find 
the  following  special  values  of  D^,  with  their  differences : 


Epoch. 

A* 

A!- 

A2. 

1750 
1800 
1850 
1900 
1950 
2000 
2050 

-0' 
-0 
0 
+  0 
0 
0 
0 

'•153  60 
•071  01 
•000  00 
•059  41 
•107  22 
•143  44 
•16805 

+  8259 
7101 
5941 
4781 
3622 
2461 
1300 

-1158 
1160 
1160 
1159 
1161 
1161 

2100 

0 

•181  05 

These  values  may  be  represented  in  the  form  : 
DI€I  =  0"-130  42  T-  0"-023  20T2. 
Then,  by  integration, 

Ae1  =  0"'0652ir2-0"-00773T3,  ................  (27) 

which  is  the  numerator  of  the  fraction. 

To  express  the  denominator  we  may  choose  either  of  two 
methods.  The  simplest  in  form,  but  not  the  shortest  numerically, 
is  to  compute  the  numerical  values  of  the  nearly  constant 

quantity  r      ,      from  the  preceding  values  of  ^\fs  for  1750,  1950, 
tan  2  V^ 

and  2100,  and  develop  them  in  the  form  a  +  fc^+cT2.  The 
product  of  this  development  into  the  above  value  of  Ae,  mul- 
tiplied by  cosec(e0  +  Ae^,  (from  which  we  may  drop  Ae^  will 


The   other   method   is   to   develop   the   denominator  and   its 
reciprocal  thus  : 


§  131.]  AUXILIARY  ANGLES  246 

We  shall,  in  form,  change  the  product 

into 


Changing   the   value  (22)  of   \fs  to  arc,  and,  for   convenience, 
expressing  the  result  in  units  of  the  sixth  place,  we  have 


40-952  +  0'0086r-  Q-QQ20T2 
^~  ~~T~~ 

The  product  of  this  into  2Aex  gives 

A  ~L_  =  5"-3409r-  0//'6320r2  -  0//'0004T3. 


Owing  to  the  minuteness  of  the  quantities  in  question,  we 
may  take  f  —  f0  as  equal  to  its  tangent,  and  neglect  Aex  in  the 
denominator  of  (24).  We  thus  obtain  the  value  of  f—  &  by 
multiplying  the  last  equation  by 

cosece0  =  2-512;   log  =  0'40002. 
This  gives 

(28) 


The  combination  of  this  expression  with  (23)  and  (25)  gives 
the  values  of  f  ,  £0,  and  z. 


-018T3   .......  (29) 


131.  Computation  of  angle  between  the  equators. 

It  only  remains  to  develop  6.  There  are  two  ways  of  doing 
this,  the  agreement  from  the  results  of  which  will  serve  as  a 
control  of  the  correctness  of  the  computation.  The  most  ready 
method  is  to  solve  the  triangle  (70P0P,  which  gives 

.    ,.     sin  e0sin  \Is 

sin  6  =  —          -  y  . 
cosf 


246  PRECESSION   AND   NUTATION  131. 

By  this  formula,  using  the  values  of  \/s  already  found,  we 
compute  the  following  values  of  0 : 


Epoch. 

e. 

A* 

Ao. 

A3. 

1750 

-  2005"-50 

+  1002" 

•84 

1800 

-  1002  -66 

1002 

•66 

-  22 

-4 

1850 

0  -00 

1002 

•44 

-  94- 

-2 

1900 

+  1002  -44 

1002 

•20 

—  ^ITT 

-  27 

-3 

1950 

2004  -64 

1001 

•93 

Zif  1 

-4 

2000 

3006  -57 

1001 

•62 

" 

-3 

2050 

4008  -19 

1001 

•28 

-34 

2100 

5009  -47 

These  values  give  the  development 

0  =  2005//-llT-()//-43T2--0"-04ir3 (30) 

The  other  method  of  developing  6  is  from  the  equation  (18), 

ivhich  gives  ^Q 

—  =  n  cos  Z  —  ^P  sin  2e  cos  z,    (31) 

which  will  lead  to  a  result  in  agreement  with  the  above. 

By  interpolation  of  the  various  quantities  we  have  tabulated, 
the  position  of  the  equator  and  equinox  at  any  epoch  between 
1750  and  2100  is  found  relatively  to  the  positions  for  1850  as 
the  initial  epoch.  A  similar  computation  may  be  made,  using 
1900,  or  other  epochs,  as  the  initial  one.  The  results  of  such 
computations  are  tabulated  in  Appendix  IV. 


Section  III.    Nutation. 

132.  Motion  of  nutation. 

The  pole  P,  whose  motion  has  just  been  developed,  is  the 
mean,  not  the  actual  pole.  The  latter  moves  round  the  mean 
pole  with  the  motion  called  nutation,  which  arises  in  the  following 
way: 

When  the  theoretical  expressions  for  the  motion  of  the  pole 
under  the  combined  action  of  the  sun  and  moon  are  derived 


132.] 


MOTION  OF  NUTATION 


247 


from  the  theory  of  rotating  bodies,  it  is  found  that  this  motion 
may  be  resolved  into  two,  one  depending  on  the  longitudes  of 
the  sun  and  moon,  and  therefore  periodic ;  the  other  independent 
of  these  longitudes,  and  therefore  progressive.  Each  term 
expressive  of  a  periodic  motion,  if  taken  separately,  would  bring 
the  pole  back  to  its  original  position  at  the  end  of  a  revolution 
of  the  sun  or  moon.  In  fact  the  largest  component  of  this 
motion  goes  through  two  periods  in  one  such  revolution. 

--r -R 


FIG.  27. 

The  progressive  motion  includes  that  of  precession,  described 
in  the  two  preceding  sections.  But  a  part  of  this  motion  is 
dependent  upon  the  longitude  of  the  moon's  node,  which  makes 
a  revolution  in  J8'6  years.  This  part  of  the  motion  being 
periodic,  is  included  in  the  nutation,  and  in  fact  forms  much  the 
largest  term  of  the  nutation. 

To  show  the  relation  of  these  terms  to  precession,  let  P  and  G 
be  the  respective  poles  of  the  equator  and  ecliptic,  and  M  that 
of  the  moon's  orbit.  Then  the  progressive  motions  produced  by 


248  PEECESSION   AND   NUTATION  [§  132, 

the  two  bodies  are  in  the  respective  directions  PE  and  PF,  at 
right  angles  to  PC  and  PM  respectively.  The  lunar  component 
PF  is  more  than  double  the  other.  The  resultant  of  the  two  is 
an  instantaneous  motion  PR,  making  an  angle  with  PE,  which 
depends  on  the  position  of  the  pole  M  relative  to  P  and  C. 

Now  M  revolves  round  C  at  a  nearly  invariable  distance  of 
5°H',  in  a  period  of  18'6  years.  The  result  is  that  the  direction 
PR  of  the  motion  of  P  continually  oscillates  on  one  side  and  the 
other  of  PE  in  a  period  equal  to  that  of  the  moon's  node,  and 
P  itself  described  a  sinuous  curve  PS  on  the  celestial  sphere. 
The  motion  on  this  curve  is  resolved  into  two  components,  the 
one  always  at  right  angles  to  PC,  being  that  of  the  mean  pole, 
the  other  a  revolution  round  the  mean  pole  in  a  period  of 
18*6  years. 

The  former  is  the  luni-solar  precession.  The  pole  C  being  both 
that  of  the  ecliptic  and  the  centre  around  which  the  pole  of  the 
moon's  orbit  revolves  with  the  nodes,  it  follows  that  the  pre- 
cessions produced  by  the  sun  and  moon  are  combined  to  produce 
the  total  precession.  When  the  actual  pole,  as  it  describes  its- 
sinuous  curve,  is  referred  to  the  mean  pole,  it  is  found  to 
revolve  round  it  in  a  somewhat  irregular  curve,  not  very  different 
from  a  circle,  which  revolution,  as  already  stated,  is  the  nutation. 

To  show  how  the  latter  is  expressed,  let  P0  be  the  position  of 
the  mean  pole,  and  P  that  of  the  true  pole  at  any  time,  while  C? 
as  before,  is  the  pole  of  the  ecliptic.  Draw  PQQ  perpendicular 
to  OP0.  Then  the  angle  at  C  is  so  minute,  never  reaching  '20f\ 
that  we  may  regard  PQQP  as  a  right  angle,  and  the  position  of 
P  relative  to  PQ  may  then  be  expressed  by  the  rectangular 
coordinates  QPQ  and  QP.  But,  for  all  the  purposes  of  astronomy, 
it  is  convenient  to  use,  instead  of  PQQ,  the  angle  P0CQ,  con- 
nected with  it  by  the  relation 


The  maximum  value  of  the  distance  P0P  between  the  poles  is- 
about  10",  a  quantity  so  small  in  comparison  with  e  that  their 
ratio  may  be  treated  as  an  infinitesimal,  and  the  triangle  P0QP 
regarded  as  plane.  Under  this  restriction  the  component  QP  of 
the  nutation  will  be  the  change  in  the  obliquity,  and  P0CQ  the 


§  133.] 


MOTION   OF  NUTATION 


249' 


change  in  the  luni-solar  precession,  produced  by  the  nutation. 
They  are  represented  by  Ae  and  Ai/r,  thus  according  with  the 
notation  of  the  last  chapter. 

The  term  of  nutation  depending  on  the  moon's  node  is  more 
than  12  times  as  large  as  the  largest  term  depending  on  the 
sun's  longitude,  and  more  than  50  times  as  large  as  the  largest 
depending  on  the  moon's  longitude.  Its  effects  on  obliquity  and 
precession  at  the  epoch  1900  are  : 

Ae  =  9"-21  cos  Q 

' 


Q  being  the  longitude  of  the  moon's  node. 

Owing  to  the  secular  diminution  of  the  obliquity  these  terms 
are  effected  by  a  minute  secular  variation,  which  need  not  be 
considered  at  present. 

PO 

C- 


FIG.  28. 

The  coefficient  9"*21  of  cos  £}  in  the  expression  for  e  is  called 
the  constant  of  nutation.  The  value  here  assigned  is  the  mean 
result  of  all  available  observations  up  to  1895.* 

133.  Theoretical  relations  of  precession  and  nutation. 

In  presenting  the  results  of  the  theory  of  the  relation  between 
the  motions  of  precession  and  nutation,  the  figure  of  .the  earth 
and  the  mass  of  the  moon,  I  adopt  the  constant  numerical 
coefficients  from  Oppolzer's  exhaustive  investigation.!  We  put 

C,  the  moment  of  inertia  of  the  earth  around  the  axis  of 
rotation. 


*  Elements  and  Constants,  pp.  130,  189,  and  195.  The  results  of  a  subsequent 
adjustment  of  the  mass  of  the  moon  led  to  the  theoretical  value  9""214,  which, 
however,  was  not  accepted  by  the  Paris  Conference  of  1896. 

t  Lehrbuch  zur  Bahnbestimmung  der  Planeten  und  Cometen,  2nd  edition,  Leipzig,. 
1882.  See  also  Elements  and  Constants,  §67,  p.  131. 


250  PRECESSION   AND  NUTATION  [§  133. 

A,  the  mean  of  the  moments  around  an  axis  perpendicular 
to  that  of  rotation. 

/UL,  the  ratio  of  the  mass  of  the  moon  to  that  of  the  earth. 

N't  the  constant  of  nutation,  as  above  defined. 

Pv  that  part  of  the  luni-solar  precession  in  a  solar  year  of 
365-2422  days  which  is  due  to  the  action  of  the  moon. 

PI,  that  part  of  the  same  precession  which  is  due  to  the  action 
of  the  sun. 

H,  K,  K',  numerical  coefficients,  functions  of  the  elements  of 
the  earth's  orbit  round  the  sun,  the  moon's  orbit  round  the  earth, 
and,  in  the  case  of  K',  of  the  volume  of  the  earth  and  the 
intensity  of  its  gravity. 

We  then  have  the  following  expressions  for  N,  Px,  and  P/  : 


C—A 


.(33) 


Pf  TT-f  (j  —  A 

^=K  cose  —  ^j— 

O  ^ 

If  we  express  N,  Pp  and  P/  in  seconds  of  arc,  the  logarithms 
of  H,  K,  and  K'  as  computed  from  gravitational  theory  are  : 

log  H  =  5-40289   \ 

log  K  =  5-975043  1  ......................  (34) 

log  JT  =  372508  J 

The  expression  given  by  theory  for  the  precessional  constant 
of  the  preceding  chapter  is  found  by  a  comparison  with  (2) 
and  (3).  Since 


we  have 


_ 

If  —  —  —  ,  or  the  mechanical  ellipticity  of  the  earth,  and  /UL,  the 

6 

mass  of  the  moon,  were  known,  the  value  of  P  could  by  means 
of  the  preceding  equation  be  determined  by  theory,  as  could  also 
that  of  N.  But,  as  neither  of  these  quantities  has  yet  been 


§  134.]  THEORETICAL  RELATIONS  251 

determined   with  the  precision  necessary  for  this,   it   is    more 

Q ^ 

common  to  determine  /x,  and  — — —    from    the    equations    (33) 

G 

>and  (34)  by  means  of  the  observed  values  of  N  and  p. 
Let  us  then  take,  as  unknown  quantities, 


C-A 

y=-r- 

We  then  have  the  equations 


*  54^9066, 


From  the  last  equation  we  derive 

log  &?/  =  5-598  84  -10, 


=  0-003  280  57, 
=  1-5-82-62, 


134.  Numerical  expression  of  the  nutation. 

When  the  theory  of  the  motion  of  the  pole  is  completely 
'developed,  it  is  found  that  the  values  of  A\/^  and  Ae  are  expressed 
as  the  sum  of  an  infinite  series  of  terms,  each  consisting  of  a 
coefficient  into  the  sine  or  cosine  of  some  combination  of  the 
following  angles  : 

ft,  the  longitude  of  the  moon's  node  ; 

L,  the  sun's  mean  longitude  ; 

TT',  the  longitude  of  the  sun's  perigee  ; 

([,  the  moon's  mean  longitude  ; 

(/,  the  moon's  mean  anomaly  : 

D^((-L. 

By  referring  to  Appendix  VII.,  it  will  be  seen  that  several 
of  the  terms  there  given  have  coefficients  of  the  order  of 
magnitude  0"'01,  or  less.  These  are  commonly  ignored  as 


252  PKECESSION  AND   NUTATION  [§134. 

unimportant.    The  only  terms  in  Ai//-  and  Ae  used  in  practice  are 
the  following  : 

Terms  depending  on  the  moon's  node. 

A^=-l7"-234    sin    0,        Ae=     9"'210cos    0, 
-   0  -OlTTsin    O, 
+   0-209    sin  20.  -  0"'090  cos  20. 

Terms  depending  on  the  sun's  longitude. 

-  l"-272  sin  2£,  Ae  =  +  0"'551  cos  2L, 

+  0-126  sin  (L  -  *'),  +  0  '022  cos  (3Z  -  TT'), 

-  0  -050  sin  (3Z  -  TT'),  -  0  '009  cos  (L  +  TT').. 
+  0 


Terms  depending  on  the  moons  longitude. 
=  -0"'204sin2([,  Ae=  +0//'089cos2d, 

+  0  -068  sin  g,  +0  '018  cos(2((-Q), 

-0  -034sin(2((-Q),  -0  '005  cos  (2  <(-g\ 

-0  -0268^(2(1+^), 

+  0  -Oil  sin  (2(-0),  +0  -Ollcos(2((+^).. 

+  0  -Olosin^D-g), 
+  0  -006  sin  2D. 

The  terms  dependent  on  the  sun's  longitude  are  sometimes. 
modified  by  using  the  true  longitude  instead  of  the  mean  as- 
the  argument,  thus  assimilating  them  with  the  aberration. 

Putting 

0,  the  sun's  true  longitude, 

we  have,  to  terms  of  the  first  order  in  the  eccentricity, 

L  =  0  —  2e  sin  (  0  —  TT'), 
2£  =  20  -4esin(0-Tr/), 
sin  2L  =  sin  2  0  —  4>e  sin  (  ©  —  TT')  cos  2  O  , 


-2ecos(30  -  TT')  +  2e  cos  (  O  +  TT')- 

In  all  the  terms  except  the  first  we  may  use   O  for  L,  the. 
error  thus  arising  being  only  0"'002.     We  have 


§  134.]     NUMERICAL  EXPRESSION  OF  THE   NUTATION          253 

Multiplying  the  above  value  of  sin  '2L  by  1"'272,  and  com- 
bining the  smaller  terms  of  the  product  with  the  corresponding 
ones  of  the  original  expression,  we  find,  for  the  sun-terms, 

A\/r=-l"-272sin20, 

+  0  -126  sin  (©-^'X 

-0  •007sin(30-7r/), 
-0  •022sin(O+7r/). 

Substituting  for  TT'  its  numerical  value  for  1900,  which  we 
may  take  as  a  constant,  /  =  281°'2 

and  combining  the  second  and  fourth  terms  into  one,  we  find 


+  0  -147  sin (0+82°), 
-0  -007  sin  (.3  0+79). 

By  a  similar  process  we  find,  for  the  nutation  of  the  obliquity, 

Ae=+0"'551cos20, 

+  0  '009  cos  (   0-79°), 
+  0  -004008(30+79°). 

NOTES  AND    REFERENCES  TO    PRECESSION   AND   NUTATION. 

The  fact  of  the  precession  of  the  equinoxes  was  originally  discovered  by 
Hipparchus,  who  nourished  during  the  second  century  before  the  Christian 
era.  His  method  was  the  same  in  principle  as  that  adopted  in  our  time 
for  determining  the  position  of  the  equinoxes  among  the  stars.  It  was  seen 
by  the  astronomers  of  ancient  times,  that  there  were  two  methods  of 
determining  the  length  of  the  year :  the  one  by  the  interval  between  the 
times  of  the  equinoxes  ;  the  other  by  the  time  of  one  apparent  revolution 
of  the  sun  among  the  stars.  As,  owing  to  the  invisibility  of  the  stars  when 
the  sun  was  above  the  horizon,  it  was  impossible  to  compare  the  sun  and 
stars  directly,  an  indirect  method  was  adopted,  using  the  position  of  the 
moon  as  an  intermediate  point  of  reference.  At  the  middle  of  a  total 
eclipse  of  the  moon,  the  latter  was  known  to  be  directly  opposite  the  sun, 
and  its  position  among  the  stars  could  be  determined.  The  distance  of  the 
moon  from  the  sun  could  also  be  measured  before  sunset,  and  from  a  star 
after  sunset. 

The  times  of  the  equinoxes  were  those  when  sunset  was  exactly  opposite 
to  sunrise.  By  comparing  the  results  of  its  own  observations  with  those  of 
his  predecessors,  it  was  found  by  Hipparchus  that  the  position  of  the  stars 


254  PEECESSION  AND   NUTATION 

Spica  and  Eegulus  had  changed  relative  to  the  equinoxes,  and  that,  while 
the  solar  year  as  determined  by  the  equinoxes  was  several  minutes  less  than 
365^  days,  the  time  of  revolution  among  the  stars  was  several  minutes 
greater.  His  estimate  of  the  motion  was,  however,  materially  too  small, 
being  1°  in  a  hundred  years  instead  of  1°  in  70  years,  which  we  now  know 
to  be  the  case. 

In  our  time  the  precessional  motion  is  determined  in  two  ways  :  from  the 
observed  declinations  of  the  stars  and  from  their  observed  E.A.'s.  Owing 
to  the  continual  motion  of  the  pole  toward  the  position  of  the  equinox,  the 
stars  situated  in  less  than  90°  of  E.A.  on  either  side  of  the  equinoxes  are 
continually  increasing  in  declination  ;  while  those  in  the  opposite  quarter 
of  the  heavens  are  diminishing.  Hence  by  comparing  the  observed  declina- 
tions of  the  stars  at  epochs  as  distant  as  possible,  the  amount  of  the  annual 
motion  of  the  pole,  or  the  value  of  the  speed  ?i,  can  be  determined. 

The  actual  E.A.'s  of  the  stars  being  determined  by  comparison  with  the 
sun  on  a  system  which  will  be  explained  in  a  subsequent  chapter,  the  actual 
motion  of  the  equinoxes  along  the  equator,  or  the  value  of  m,  can  also  be 
determined  by  observation.  The  actual  amount  of  the  motion  is  inferred 
from  a  combination  of  the  two  methods,  taking  the  means  which  seem  most 
probable  when  due  allowance  is  made  for  the  various  sources  of  error  to 
which  each  is  subject. 

That  component  of  the  motion  of  the  ecliptic  which  takes  place  around 
the  line  of  the  equinoxes  as  an  axis,  or  the  value  of  K  cos  iV,  can  be  deter- 
mined by  observations  of  the  obliquity  at  various  epochs,  but  the  component 
at  right  angles  to  this,  on  which  alone  the  motion  of  the  equinox  depends, 
and  which  enters  through  the  quantity  A,  cannot  be  accurately  determined 
by  direct  observation. 

The  motion  of  the  ecliptic  can  also  be  determined  theoretically  from  the 
action  of  the  planets,  when  the  masses  of  the  latter  are  known.  Our 
knowledge  of  these  motions  is  subject  to  correction,  and,  in  consequence, 
the  numbers  expressing  the  motion  have  also  been  corrected  from  time  to. 
time.  Two  independent  corrections,  one  to  the  motion  of  the  equator  and 
the  other  to  that  of  the  ecliptic,  have  been  made  from  time  to  time.  This 
has  been  productive  of  more  or  less  confusion  in  the  combination  of  the  two- 
numbers  so  as  to  obtain  the  definitive  values  of  the  precessional  motion. 

The  first  values  of  these  motions  which  have  been  extensively  used  in 
astronomy  were  derived  by  Bessel,  and  are  found  in  a  memoir  to  which 
a  prize  was  awarded  by  the  Berlin  Academy  of  Sciences  in  1815.  These 
values  were  afterward  corrected  by  Bessel  and  the  results  embodied  in  the 
Tabulae  Regiomontanae,  long  the  most  accurate  handbook  of  formulae, 
constants,  and  tables  relating  to  the  positions  of  the  stars.  The  values  here 
found  were,  therefore,  in  very  general  use  for  a  number  of  years. 

When  the  Poulkova  Observatory  was  founded  (1839),  one  of  the  objects 
mainly  in  view  was  the  accurate  determination  of  the  positions  of  the 


NOTES   AND   EEFERENCES  255- 

principal  fixed  stars,  and  of  the  constants  pertaining  to  their  reduction. 
This  led  to  an  investigation  of  the  constant  of  precession  by  Otto  Struve, 
which  was  published  by  the  St,  Petersburg  Academy  of  Science  in  1843, 
and  was  based  upon  a  comparison  of  the  observations  of  Bradley,  1750-1755, 
with  those  of  Bessel.  The  motions  found  by  Struve  were  modified  by 
Peters,  and  are  markedly  larger  than  those  of  Bessel.  They  gradually 
replaced  the  latter,  their  use  becoming  general  through  the  last  half  of  the 
19th  century. 

Values  were  also  derived  by  Leverrier  and  slightly  modified  by  Oppolzer. 
The  following  is  a  list  of  the  motions  just  mentioned  : 

Annual  precessional  motions  by  various  authorities  for  the  epoch  1850. 

General 
Precession. 

Bessel         -         -         -  50"'2357  46"'0591  20"'0547 

Struve-Peters    -         -  50  '2522  46  '0764  20  '0564 

Leverrier  *  -  50  "2357  46  '0601  20  '0524 

Oppolzer    ---  50  '2346  46  "0593  20  '0515 

Values  of  the  precession  have  also  been  derived  by  LUDWIG  STRUVE, 
BOLTE,  DRETER  (in  the  journal  Copernicus,  vol.  ii.,  Dublin,  1882),  and 
NYRE*N. 

PETERS,  C.  A.  F.,  Numerus  Constans  Nutationis,  published  by  the  St. 
Petersburg  Academy  of  Sciences,  contains  the  first  thorough  development 
of  the  theory  of  Nutation. 

OPPOLZER,  Lehrbuch  zur  Bahnbestimmung ,  Leipzig,  1882,  contains  an  ex- 
tended mathematical  theory  of  precession  and  nutation  ;  his  numerical  values 
are  based  on  those  of  Leverrier. 

In  1896  a  conference  of  the  directors  of  the  National  Ephemerides  of 
England,  France,  Germany,  and  the  United  States  was  held  at  Paris  for 
the  purpose  of  deciding  upon  a  uniform  set  of  astronomical  constants,  and 
a  system  of  star-reductions  to  be  adopted  in  the  several  publications. 
Values  of  the  precessional  motions  were  also  worked  out  for  this  purpose 
during  the  following  year,  in  a  research  found  in  Astronomical  Papers  of  the 
American  Ephemeris,  vol.  viii.  At  that  time  the  reinvestigation  of  the 
motions  and  elements  of  the  planets  had  just  been  completed,  which  led  to  a, 
more  rigorous  determination  of  the  motion  of  the  ecliptic.  The  resulting 
precessional  motions  are  those  used  in  the  present  work,  and  mostly  adopted 
in  the  national  ephemerides  since  1901. 

The  Elements  of  the  four  inner  Planets  and  the  Fundamental  Constants  of 
Astronomy,  published  by  the  American  Nautical  Almanac  in  1895,  also 
gives  values  of  the  precessional  motions.  But  these  were  superseded  by 
the  value  of  1897,  which  had  not  then  been  worked  out. 


*For  the  Julian,  not  the  solar  year. 


256  PRECESSION   AND  NUTATION 

The  amount  of  the  nutation,  computed  from  formulae  substantially 
identical  with  those  just  given,  is  found  in  all  the  national  ephemerides  for 
intervals  admitting  of  convenient  interpolation  to  any  date.  The  periods 
of  the  terms  depending  on  the  moon's  node,  and  on  the  sun's  longitude,  are 
so  long  that  an  ephemeris  for  every  tenth  day  will  suffice  so  far  as  they 
Are  concerned.  Those  depending  on  the  moon's  longitude,  if  used  at  all, 
must  be  given  for  every  day.  As  for  many  astronomical  purposes,  it  is 
sufficient  and  sometimes  even  desirable  to  omit  the  small  terms  in  the 
nutation,  the  two  classes  of  terms  are  given  separately  in  the  American 
Ephemeris  and  in  the  British  Nautical  Almanac.  In  the  former  the 
ephemeris  of  the  larger  terms  is  given  for  every  ten  days  only  ;  and  that  in 
two  separate  tables, — one  containing  the  values  computed  from  the  old 
constants  of  Struve  and  Peters,  the  other  from  the  present  adopted  values. 
A  separate  table  is  given  of  the  small  terms  for  every  day.  In  the  British 
Ephemeris  both  classes  of  terms  are  given  for  every  day.  In  the 
Connaissance  des  Temps  the  complete  nutation  is  given  for  every  day.  The 
reason  for  tabulating  the  two  classes  of  terms  separately  will  be  set  forth  in 
the  chapter  on  the  reduction  of  apparent  places  of  the  fixed  stars. 

For  convenience  the  following  values  of  the  term  of  the  nutation  de- 
pending on  the  longitude  of  the  node  are  given. 

A^.  Ae. 

BESSEL,  Tabulae  Regiomontanae,     -16"'783sinft  8"'977cosi2 

PETERS,  Numerus  constans,  -17  '258  9  '224 

OPPOLZER,  Bahnbestimmung,  - 17  '274  9  '236 


PART  III. 

REDUCTION  AND   DETERMINATION   OF 
POSITIONS   OF  THE   FIXED    STARS. 


N.S.A. 


CHAPTER  X. 

EEDUCTION   OF   MEAN  PLACES   OF   THE   FIXED   STARS 
FROM  ONE  EPOCH  TO  ANOTHER. 

135.  The  mean  place  of  a  fixed  star  at  any  time  is  its 
apparent  position  on  the  celestial  sphere,  as  it  would  be  seen 
by  an  observer  at  rest  on  the  sun.  It  is  commonly  expressed 
by  coordinates  referred  to  the  mean  pole  and  equinox  of  the 
beginning  of  some  year. 

The  apparent  place  of  such  a  star  is  its  position  on  the  sphere 
as  it  is  actually  seen  by  an  observer  on  the  moving  earth, 
referred  to  the  actual  pole  and  equinox  of  the  date. 

The  problem  of  the  reduction  of  places  of  the  fixed  stars  is 
that  of  determining  the  apparent  place  at  one  time  from  the 
mean  place  at  another,  or  vice  versa.  It  involves  correction  for 
the  effects  of  the  following  causes : 

1.  The  proper  motion  of  the  star  between  the  two  epochs ; 

2.  Precession; 

3.  Nutation; 

4.  Aberration; 

5.  Annual  Parallax. 

Of  these  causes  1,  4,  and  5  change  the  actual  direction  in 
which  the  star  is  seen,  while  2  and  3  change  only  the  axes  of 
coordinates,  without  affecting  the  actual  direction  of  the  star. 

The  reduction  involves  two  steps.  Firstly,  the  mean  place 
of  the  star  is  reduced  from  one  epoch  to  the  other  by  applying 


260  MEAN  PLACES  OF   THE   FIXED   STARS  [§135. 

the  effects  of  precession  and  proper  motion.  Secondly,  this 
mean  place  is  corrected  for  the  effect  of  nutation,  aberration, 
and  parallax. 

The  correction  for  parallax  is  generally  neglected  in  ephe- 
merides,  owing  to  its  minuteness,  and  the  lack,  except  in  a 
few  cases,  of  exact  knowledge  of  its  value. 

The  reduction  of  the  mean  place  from  one  epoch  to  another 
is  so  far  distinct  from  the  computation  of  the  apparent  place  that 
we  consider  them  in  separate  chapters.  In  the  present  chapter 
the  theory  of  mean  places,  as  affected  by  proper  motion  and 
precession,  will  be  studied. 

In  practice  mean  places  are  always  referred  to  the  equator 
and  equinox  of  the  beginning  of  some  solar  year.  Hence  the 
reduction  for  precession  is  always  made  for  an  integral  number 
of  years. 


Section  I.    The  Proper  Motions  of  the  Stars. 

136.  Each  fixed  star  is,  with  very  few  exceptions,  so  widely 
separated  from  all  others  that,  during  our  time,  and  perhaps 


FIG.  29. 

for  ages  to  come,  it  may  be  supposed  to  move  in  a  straight  line 
with  uniform  velocity.  In  cases  where  this  is  not  true,  owing 
to  the  star  being  a  member  of  a  system,  it  is  still  true  of  the 
centre  of  mass  of  the  system.  What  we  have  therefore  to 
consider  is  the  effect  of  the  uniform  rectilinear  motion  of  a  star 
in  space  upon  its  apparent  motion  on  the  celestial  sphere. 


§  136.]  THE   PROPER   MOTIONS   OF  THE   STARS  261 

Let  E  be  the  position  of  the  earth,  or  more  exactly  that  of 
the  sun,  this  being  the  mean  position  of  the  earth  during  any 
one  year,  and  therefore  the  actual  origin  to  which  the  mean 
positions  and  motions  of  the  stars  are  referred. 

Let  the  star  be  moving  relatively  to  the  sun  in  the  direction 
SSl  with  the  uniform  linear  speed  v,  and  let  us  put 

r,  the  distance  ES  of  the  star,  called  also  the  radial  line  ; 

W,  the  angle  ESSl  which  the  direction  of  the  motion  makes 
with  the  radial  line  ; 

p,  the  speed  with  which  r  is  increasing,  called  the  radial 
velocity  ; 

TT,  the  annual  parallax  of  the  star  ; 

/UL,  the  angular  speed  of  its  apparent  motion  on  the  celestial 
sphere,  as  seen  from  the  sun.  We  then  have 

dW 


dr 


We  now  have  to  find  the  variation  of  //.  with  the  time. 
By  differentiation  of  the  first  equation  and  simple  reductions, 
we  find  o 


Substituting  for  sin  W  and  cos  W  their  values  from  (1),  this 
equation  reduces  to 


Instead  of  r,  the  distance  of  the  star,  we  use  its  annual 
parallax,  connected  with  a,  the  mean  distance  of  the  earth 
from  the  sun,  by  the  relation 

a  =  r  sin  TT. 

Using  a  as  the  astronomical  unit  of  length,  we  should  define 
v  in  terms  of  this  unit,  using  it  to  indicate  radii  of  the  earth's 

orbit  in  unit  of  time.     Doing  this  and  substituting  sin  ?r  for  - 
we  find 


262  MEAN  PLACES   OF   THE   FIXED   STARS  [§  136. 

In  this  equation  /u.  is  expressed  in  circular  measure.  In 
astronomical  practice  it  is  common  to  express  jm  in  seconds  of 
arc  per  year,  and  the  radial  velocity  in  kilometres  per  second. 
The  value  of  p  to  be  used  in  the  last  equation  is  found  by 
dividing  the  speed  in  kilometres  per  second  by  4'75.* 

If  we  put  /*"  for  the  annual  motion  in  seconds  of  arc,  and  IT" 
for  the  parallax  in  seconds  of  arc,  we  shall  have 

/x  =  //'  sin  1"  } 

sin  TT  =  TT"  sin  V  \ (3) 

Dtfj."  =  —  2/i'V/o  sin  1"J 

The  radial  speed  p  can  be  measured  only  with  the  spectro- 
scope, and  is  known  only  for  a  few  hundred  of  the  brighter 
stars.  Among  the  stars  whose  radial  speed  and  parallax  have 
both  been  determined,  1830  Groombridge  is  that  which  will 
give  much  the  largest  value  of  this  change.  The  measures  of 
its  radial  speed  at  the  Lick  Observatory  give 

P--20. 

For  it  we  have  also 

//          ty// 
/*      =   * 

and,  with  much  uncertainty, 

7r"  =  0"14, 
and  thus,  for  1830  Groombridge, 

iV"=+o"-oooi9. 

This  change  is  too  small  to  be  detected  until  accurate  observa- 
tions shall  have  extended  through  fully  a  century :  and  as  it  is 
exceptionally  large,  the  consideration  of  the  change  in  the  case 
of  the  stars  in  general  belongs  to  the  astronomy  of  the  future. 
In  the  present  state  of  astronomy  we  may,  therefore,  assume 
that  by  its  proper  motion  each  star  moves  on  a  great  circle 
with  an  invariable  angular  speed.  We  put 

/UL,  this  constant  angular  speed; 


*  This  factor  is  connected  with  the  solar  parallax  by  the  relation 

[1-62003] 

Factor = -7^r*- ^— J 

O  s  par.  in  sees. 

The  value  as  given  therefore  corresponds  to  par.  =  8"*776.     For  the  value 
•8"'80,  still  in  common  use,  we  have  4*7375  for  the  divisor. 


§  137.]  THE   PROPER  MOTIONS  OF  THE   STARS  263 

iV,  the  angle  which  its  direction  makes  with  the  meridian  of 
the  star,  counted  from  North  toward  East.  We  then  have,  for 
the  rates  of  change  in  R.A.  and  Dec., 

Proper  motion  in  R.  A.,       M«  =  M  si*1  N  sec  S\  ,^ 

Proper  motion  in  Dec.,      /x$  =  /UL  cos  N         j 

137.  Reduction  for  proper  motion. 

The  mean  place  of  a  star  at  any  epoch  is  not  necessarily 
referred  to  the  equator  and  equinox  of  that  epoch.  We  may 
have  occasion  to  refer  it  to  the  coordinate  axes  of  any  other 
epoch.  It  follows  that  the  reduction  for  proper 
motion  is  quite  distinct  from  that  for  precession. 
We  therefore  begin  by  finding  the  effect  of  proper 
motion  when  the  axes  of  reference  remain  fixed. 
We  put 

a0,  <J0,  N0,  the  coordinates  of  the  star  and  the 
direction  of  its  proper  motion  at  the  initial  epoch  ; 

OL,  S,  N,  the  corresponding  quantities  for  an  epoch 
later  by  the  time  t,  referred  to  the  same  equator 
and  equinox. 

In  Fig.  30  let  P  be  the  pole.  During  the  interval  t  the  star 
will  have  moved  over  an  arc 


so  that  PSS'  is  a  triangle  in  which 

Angle  S  =  N0,  for  the  initial  date  ; 
Exterior  Angle  S'  =  N,  for  the  final  date. 

The  equations  between  the  parts  of  this  triangle  enable  us  to 
determine  oc,  S,  and  N  by  the  rigorous  equations 
cos  8  sin  (a.  —  a0)  =  sin  NQ  sin  ^t, 
cos  (5  cos  (OL  —  GCO)  =  cos  $0  cos  jj.t  —  sin  SQ  cos  N0  sin  jj.tt 

sin  S  =  sin  S0  cos  jmt  +  cos  SQ  cos  N0  sin  /*£,  ...............  (5) 

cos  S  sin  N  =  cos  S0  sin  NQ, 

cos  8  cos  N  =  cos  $0  cos  NQ  cos  ^t  —  sin  <S0  sin  /mt  ................  (6) 

The  use  of  these  equations  in  their  rigorous  form  can  be 
necessary  only  in  the  rare  case,  which  has  not  yet  occurred  in 
practice,  when  the  proper  motion  is  an  important  fraction  of  the 


264  MEAN   PLACES   OF  THE   FIXED   STARS  [§  137. 

polar  distance  of  the  star.  In  all  ordinary  cases  a  development 
in  powers  of  the  time  to  f2  will  suffice,  and  even  this  last  term 
will  rarely  be  appreciable. 

To  effect  this  development  we  suppose  SS'  infinitesimal.     We 
then  have,  from  the  triangle  PSS', 


(7) 
dN=  sin  Sda.  =  /u.  sin  N  tan  Sdt 

Then,  by  differentiating  (4),  substituting  and  reducing, 

1 


The  derivatives  are  enclosed  in  parentheses  to  distinguish 
them  from  the  total  variations  when  precession  is  included. 
Their  value  should  satisfy  the  condition  that  the  derivative  of 


shall  vanish,  which  we  find  to  be  the  case  by  differentiating  and 
substituting  from  (8)  and  (4). 

The  preceding  equations  presuppose  that  /u.  is  expressed  in 
circular  measure.  To  transform  them  into  the  usual  notation 
of  seconds  of  arc,  we  must,  in  one  of  the  factors  of  the  second 
member,  replace  /x  by  /m  sin  V.  Then 

=  2jua/x5  sin  1"  tan  S         ) 
=  -  Ma2  sin  1"  sin  8  cos  8  )' 
which  holds  true  when  /ma  and  /ms  are  expressed  in  seconds  of  arc. 
By  Taylor's  theorem  oc  and  S  are  expressed  in  the  form 


where  the  quantities  in  the  second  member  are  the  values  for 
the  initial  epoch. 

Putting  A1a  =  /xa^  and  A1^  =  /X5^,  these  quantities  will  be  the 
principal  terms  of  oc—  o^  and  S  —  So  respectively.  Comparing 
with  (8'),  we  see  that  the  last  terms  of  (9)  may  be  written 

AxaA^  sin  1"  tan  <$0, 
-  J  A^2  sin  1"  sin  <S0  cos  ^ 


§  138.]  REDUCTION   FOR  PROPER  MOTION  265 

It   follows   that    we   may   write   the    reductions   for    proper 
motion  in  the  form 


oc  -  oc0  =  /xa<(l  +  A15  tan  (?0  sin  1")         j 
S  —  S0  =  ^t  —  J  AiOc2  sin  £0  cos  £0  sin  1"  J  ' 

where  A^  and  A^  are  the  values  of  ^Jt,  and  /m8t  expressed  in 
seconds  of  arc.  It  is  only  when  these  quantities  are  ex- 
ceptionally large  that  the  last  terms  will  become  sensible.  The 
equations  (10)  have  the  advantage  of  enabling  us  to  determine, 
almost  at  a  glance,  whether  the  terms  in  t2  are  sensible.  If  they 
are,  and  if  we  require  the  proper  motion  as  well  as  the  position 
for  the  second  epoch,  the  reduction  may  be  made  by  using  the 
mean  value  of  the  motion  for  the  two  epochs.  We  then  begin 
by  computing  (8),  and  then  the  changes  in  the  values  of  /u.a  and 
/X6  from  the  equations 


which,  being  applied  to  the  proper  motions  for  the  initial  epoch, 
will  give  them  for  the  final  epoch,  referred  to  the  same  equinox 
in  the  two  cases.  We  then  make  the  reduction  of  the  position 
by  the  formulae 

-  2x 


The  reduction  (12)  is  for  proper  motion  alone,  the  axes  of 
reference  remaining  fixed. 

Section  II.    Trigonometric  Reduction  for  Precession. 

138.  Rigorous  formulae  of  reduction. 

The  problem  before  us  now  is  :  Having  given  the  coordinates 
and  proper  motion  of  a  star,  referred  to  the  equator  and  equinox 
of  some  initial  date,  to  find  the  values  of  these  quantities 
referred  to  the  equator  and  equinox  of  some  other  date,  the 
absolute  position  and  rate  of  motion  remaining  unchanged. 

There  are  two  ways  of  solving  this  problem  :  one  by  a 
rigorous  trigonometric  computation  ;  the  other  by  a  development 
in  the  powers  of  the  time.  We  begin  with  the  first. 


266 


MEAN   PLACES  OF  THE   FIXED   STARS 


[§  138. 


Let  P0  and  P  (Fig.  31)  be  the  positions  of  the  poles  at  the  two 
epochs,  and  S  that  of  the  star.  Draw  the  arcs  PQE0,  and  PE 
from  the  poles  toward  the  respective  equinoxes.  Comparing 
this  figure  with  Figure  26,  we  see  that 

Angle  PPoffo-fi,, 
Angle  P0P#=  180°  -0, 

where  £0  and  z  are  the  angles  whose  values  have  been  developed 
in  Chapter  IX,  §§  130,  131. 
We  have  also 

Angle  SP0E0  =  R. A.  of  star  at  initial  epoch ; 
Angle  SPE  =  R. A.  at  terminal  epoch. 


FIG.  31. 

Let  (Xo,  $0,  ]ULa,  and  ^5°  be  the  coordinates  and  proper  motions 
of  the  star  when  referred  to  the  initial  equinox  ;  and 

a,  S,  ytxa,  and  /x5  the  same  quantities  referred  to  the  final  equinox. 
In  the  triangle  SPP0  we  have 

Angle  at  P0  =  o^-f  £>  =  a 

Angle  at  P  =  180°- (a -2)  =  180° -a' 

Side      P0P  =  0  !- (13) 

Side       P0S  =  90°-<$0 
Side        P/S  =  90°-5 

Of  these  parts  the  second  and  last  are  determined  from  the 
other  three  given  parts.  They  may  be  found  by  the  usual 
trigonometric  equations 


§138.]  RIGOROUS   FORMULAE  OF  REDUCTION  267 

cos  8  sin  a'  =  cos  80  sin  a  \ 

cos  8  cos  a'  =  cos  0  cos  S0  cos  a—  sin  0  sin  <5()  V  ..........  (14) 

sin  8  =  sin  0  cos  S0  cos  a  +  cos  0  sin  80j 

These  equations  give  the  shortest  computation  so  far  as  the 
number  of:  quantities  to  be  used  is  concerned.  They  may  be 
adopted  in  the  case  of  a  star  very  near  the  pole.  But  their  use 
requires  7  -place  logarithms  ;  and  in  all  ordinary  cases  they  may, 
owing  to  the  smallness  of  0,  be  advantageously  transformed  into 
others  which  will  give  a  —a  and  8'  —  8  in  terms  of  the  given 
data,  and  will  require  fewer  figures  in  the  logarithms.  To  do 
this  we  multiply  the  first  by  cos  a  and  the  second  by  sin  a  and 
subtract.  Then  we  multiply  the  first  by  sin  a  and  the  second 
by  cos  a  and  add.  The  quotient  of  the  two  results  gives  us  an 
equation  which,  after  dividing  both  terms  of  the  fraction  by 
cos  8,  we  may  write 

N 
tan(a'  —  a)  =  -g,    ..........................  (15) 

where  N  =  (cos  a  —  cos  0  cos  a  +  sin  0  tan  80)  sin  a, 

D  =  cos  0  cos2a  +  sin2a  —  sin  0  cos  a  tan  8Q. 

The  coefficient  of  sin  a  in  the  numerator  readily  reduces  to 

sm0tan<?0+2sin2£0cosa==p,  ..................  (16) 

and  the  denominator  reduces  to 

1  —p  cos  a. 

Having  determined  a'  —  a  from  (15),  we  see  from  the  first  two 
equations  (13)  that  we  have  the  following  computations  for  the 
total  change  in  R.  A.  : 


,  ,       N       p  sin  a 
tan  (a  —  a)  =  r-*- 

1  —    cos 


where  we  put  m  =  f0  +  z. 


268  MEAN  PLACES  OF  THE   FIXED  STARS  [§  138. 

The  reduction  in  declination  is  equal  to  the  difference  of  the 
sides  P0$  and  PS,  for  which  Napier's  analogy,  or  the  quotient 
of  two  of  the  Gaussian  equations  for  the  spherical  triangle, 

nl*  .............  (19) 


139.  Geometric  signification  of  the  constants. 

The  geometric  signification  of  the  arcs  a  —  a,  £0  and  0,  whose 
sum  make  up  the  reduction,  can  be  more  readily  seen  by  a 
study  of  their  relation  to  the  equator  than  by  the  construction 
we  have  used.  Let  EQQ  and  EQ  be  the  two  equators,  inter- 


FIG.   32. 


secting  at  Q,  and  E0  and  E  the  two  equinoxes.  We  then  see, 
by  transferring  the  measures  of  the  angles  f0  and  z  from  their 
vertices  at  the  poles  to  the  equator,  that 


Then  since  oco  =  EQR0  and  a  =  ER,  we  have 


a'-a  =  R()Q-RQ. 

140.  Approximate  formulae. 

The  preceding  rigorous  formulae  are  necessary  only  in  the 
case  of  stars  near  the  pole.  In  all  ordinary  cases  the  reduction 
may  be  simplified  by  the  following  process.  We  write  p  in 
the  form  p—p0+^p, 

where  p0  =  sin  0  tan  S0 


§  141.]  APPEOXIMATE   FOEMULAE  269 

Now  develop  tan  (a'  — a)  in  (17)  in  powers  of  A£>. 

,       ,        £>0sina        sin2J(9  sin  2a  /01 . 

tan  (a  —  a)  =  -^- \-— — ro  +  etc (21) 

1—  pQcosa     (l—p0cosa)2 

To  estimate  the  value  of  the  second  term  we  note  that,  in  a 
reduction  extending  over  a  hundred  years,  we  have,  approxi- 

mate1^'  J0  =  1002"  =  0-005, 

whence  sin2  JO  =  5"-0  =  Os<33. 

This  is  the  maximum  value  of  the  numerator  of  the  last  term 
of  (21)  for  this  particular  case.  Since  p0  is  small,  unless  the 
star  is  near  the  pole,  the  denominator  will  generally  differ  little 
from  unity.  For  a  reduction  through  100  years  approximate 
values  of  p0  or  p  are 

Dec.  =  80°;    p  =  0*057. 
„    =85°;    j9  =  0'115. 

The  equation  (21)  will,  therefore,  suffice  in  all  cases  when  the 
star  is  not  very  near  the  pole.  Its  computation  may  be 
facilitated  by  dividing  tan  (a'  — a)  into  three  parts,  using  the 
notation 


1  —  p0  cos  a 
A1a  =  sin2  J6>sin2a  • (22) 

A2a  =  L-j —         — r2  —  1 J  Axa  =  F\a 
Then 
of  —  a  —  A0a  +  Axa + A2a  —  Red.  from  arc  to  tangent  =  Aa.  . . .  (23) 

141.  Construction  of  tables  for  the  reduction. 

The  computation  of  these  quantities  is  shortened  by  the  tables 
of  Appendix  IV.,  of  which  the  construction  is  this : 

We  express  the  four  parts  of  a  in  seconds  of  time  by  dividing 
them  by  sin  1  s.  =  15  sin  1",  the  reciprocal  of  which  we  call  h,  so 

that  log  k  =  4-1 38  334. 

When,  and  only  when,  necessary  to  avoid  confusion,  we  indicate 


270  MEAN  PLACES  OF  THE   FIXED  STARS  [§  141. 

this  form  of  expression  by  a  suffix  s,  so  that  ps  means  p 
expressed  in  seconds  of  time,  or  ps  —  Jip.  We  then  have  from  (22) 

A0g.=/°"sina, 
1  —  p0  cos  a 

with  similar  expressions  formed  by  multiplying  Axa  and  A2a  by 

p.     We  then  have  ,_.. 

a  =  oCo+Aas  +  m  ........................  (24) 

The  constants  and  formulae  for  all  the  cases  which  ordinarily 
occur  are  found  in  Appendix  IV.,  which  also  contains  tables  to 
facilitate  the  reduction.  Table  XII.  of  this  Appendix  gives  the 
logarithm  of  T 

#  =  f- 

1  —  p  cos  a 

for  usual  values  of  p,  the  computation  of  log-fif  being  made 
with  p  cos  a  in  circular  measure,  but  the  argument  being  multi- 
plied by  the  factor  h,  so  as  to  be  expressed  in  seconds  of  time. 

Table  XIII.  gives  the  value  of  Axa,  the  argument  0  being 
replaced  by  the  time  elapsed  between  the  two  epochs,  to  which 
it  is  nearly  proportional. 

Table  XIY.  gives  the  factor  F,  by  which  Axa  is  multiplied  to 
find  A2a. 

>  Table  XV.  gives  the  reduction  from  the  sum  A0ctH-A1a+A2a, 
(which  is  the  tangent  of  Aa  expressed  in  seconds  of  time) 
to  Aa  itself.  It  is  always  subtractive  numerically. 

142.  Reduction  of  the  declination. 

Unless  the  motion  of  the  pole  is  an  important  fraction  of 
the  polar  distance  of  the  star,  we  may  use,  instead  of  (19),  the 
approximate  equation 

sec  JAa  ................  (25) 


143.  Failure  of  the  approximation  near  the  pole. 

The  boundary  of  the  region  within  which  the  use  of  Ap  ceases 
to  be  convenient  is  approximately  a  spherical  lemniscate  having 
the  pole  as  centre,  and  the  meridian  through  0  h.  and  12  h.  of 
R.A.  as  its  axis.  Practically  we  may  replace  this  curve  by  a 
pair  of  circles  as  shown  in  Figure  33. 


§  144.]  APPROXIMATION  NEAR  THE  POLE  271 

The  length  of  the  semi-axis  a  may  be  taken  as  1°  for  every 
10  years  of  the  interval  through  which  the  reduction  extends. 
The  limits  are,  in  general,  given  by  the  equation 

Polar  Distance  =  0°'10£  cos  a. 
The  argument  of  Table  XV.  approaches  the  tabular  limit  when 

Polar  Distance  =  0°'04£  sin  a. 

The  corresponding  limiting  curve  is  a  lemniscate  similar  to  that 
just  defined,  but  having  its  axis  at  right  angles  to  that  of  the 


FIG.  33. 

other.  It  is  shown  by  the  two  dotted  circles.  If,  owing  to  the 
position  of  the  star  being  within  the  limits  just  defined,  or  to  any 
other  reason,  the  rigorous  formulae  (17)  are  used,  the  computation 
can  still  be  facilitated  by  using  the  table  for  K. 

144.  Reduction  of  the  proper  motion. 

The  proper  motion  of  the  star  when  referred  to  the  final 
equinox  will  also  be  different  from  that  referred  to  the  initial 
equinox,  owing  to  the  change  in  the  direction  of  the  hour  circles. 
To  reduce  it  to  the  final  equinox,  let  us  again  refer  to  the  triangle 
PQPS,  formed  by  the  two  poles  and  the  star.  The  angle  N  of 
§  136,  Eq.  (4)  will  be  changed  by  the  angle  S,  so  that,  putting 


272  MEAN   PLACES   OF  THE   FIXED   STARS  [§  144. 

N',  the  value  of  N,  referred  to  the  final  equinox,  we  shall  have 


The  angle  S  may  be  computed  by  the  equation 

sin  S  =  sin  0  sin  a  sec  S  ......................  (26) 

The  proper  motions  referred  to  the  final  equinox  will  then  be 
given  by  the  equations 

,ua'  cos  S'  =  n  sin  (N+  S)  =  /JL.  sin  NcosS+p  cos  N  sin$"j 
=  MaCOS($cos£+yu5sin$          [ 
JULS  =  /UL  cos  (N+S)  —  fj.  cos  JV  cosS—jui  sin  JV  sin  S  I 
=  /ULS  cos  $  —  /xa  cos  <S  sin  £ 

In  the  preceding  process  of  reduction  we  have  commenced 
with  applying  the  proper  motion  during  the  interval  of  reduction 
so  as  to  use  for  oc0  and  S0  the  position  at  the  terminal  epoch, 
referred  to  the  initial  equator  and  equinox.  But  we  may  with 
equal  convenience  commence  with  the  reduction  for  precession. 
The  steps  of  the  process  will  then  be  : 

1.  Having  given  the  coordinates  oc0  and  S0  of  the  star  referred 
to  the  initial  equinox,  we  reduce  them  to  the  final  equinox,  the 
absolute  position  on  the  sphere  remaining  unchanged. 

2.  We  make  a  similar  reduction  of  the  instantaneous  proper 
motion,  so  as  to  reduce  it  to  the  new  direction  of  the  pole. 

3.  We  compute  the  absolute  motion  of  the  star  between  the 
two  epochs  by  reducing  the  position  obtained  by  step  1  with  the 
proper  motion  obtained  by  step  2. 

As  an  example  of  the  reduction,  we  take  the  star  1830  Groom- 
bridge,  of  which  the  position  and  centennial  proper  motion  for 
the  date  1875'0  are  : 

oc=ll  h.  45  m.  46120  s.;         yua=  +34198  s.  ^ 
a  =  176°  26'-53;  =  512"«97         L  •••  (A) 

S  =  +  .38°  36'  55"-55  ;  MS  =  -  577"  '97     J 

Assuming  that  a  5-place  table  of  sines  and  cosines  to  time  is 
not  at  hand,  we  have  reduced  oc  to  arc. 

We  call  this  position  (A),  and  we  propose  to  reduce  it  to 
1910'0,  an  interval  of  35  years. 


§  144.]  TRIGONOMETRIC  REDUCTION  273 

We  begin  by  computing  the  absolute  motion  of  the  star 
between  the  two  epochs,  supposing  no  change  of  the  equinox  of 
reference.  We  first  require  the  change  in  proper  motion,  which 
may  be  computed  by  (8')  : 

log/t*a          2-7101  log/>ta2     5-4202 


„  /**  2-7619  n  „  sin  1"  4-6856  -  10 

„  2  sin  1"  4-9866-10  „   sin  8    9-7952-10 

„  tan  8      9-9024-10  „  cos  8    9-8928-10 

0-361071  „  J9.8      9-7938-10 

=  -2"-296  =  -  0-153  s., 
-0"-622. 


Hence  in  35  years,  or  0*35  of  a  century, 

(A/n.)  =  -  0-054  s.  ;      (  A/*,)  =  -  0"'22, 
and,  for  1910, 

/xa  =  34-144  s.  ;  M5=  -578"-19,     .........  (B) 

which  motions  are  still  referred  to  the  original  axes. 

Reducing  the  position  to  1910  by  (10)  or  (11),  we  now  find 

oc=     176°  29'  31"-20^  (  . 

S=+   38   33  33  -22  J 

This  second  position,  which  we  call  (B),  is  that  of  the  star  in 
1910,  when  referred  to  the  equinox  of  1875. 

We  thus  have  two  positions  of  the  star  —  the  one  for  1875,  the 
other  for  1910  —  both  referred  to  the  equator  and  equinox  of  1875. 
We  shall  now  reduce  both  of  these  positions  to  the  equator  and 
equinox  of  1910. 

We  find  the  following  constants  of  reduction  from  the  general 
expressions  of  Appendix  IV.,  making  the  reversal  there  explained: 


log  h  sin  0  =  1-670  04, 

m  =  l  m.  47-527  s. 

=  26'52"-90, 
0=11'41"-66  = 


The  computation  is  now  the  following,  starting  in  (A)  with  the 

position  in  1875,  and  in  (B)  with  that  for  1910  : 

N.S.A.  s 


274 


MEAN   PLACES  OF  THE   FIXED  STARS 


[8  144. 


Reductions  0/1830  Groombridge  from  the  equinox  and 
equator  of  1875  to  those  of  1910. 


K.A.,  initial  equinox,  OLO 
f0,  1875  to  1900 

a 

log  tan  S0 

log  h  sin  Q 

log  cos  rt 


A. 

176°    26'-53 
13-44 

176°   39'-97 

9-90240 
1-67004 
9-999  26* 


B. 

176°  29'-52 
13-44 

176°  42'-96 

9-901  52 
1-67004 
9-999  29w 


log  K  Tab. 

log  sin  a 

log;*. 


1 -57l70?i 

0-00118 
8-76457 
1-57244 

0-33583 


l-57085w. 


-0-00117 
8-75804 
1-571  56 
0-32843 


\a 

2-167 

2-130 

\a 

-•004 

-•004 

h. 

m.          s. 

h. 

m. 

s. 

m 

0 

1   47-527 

0 

1 

47-527 

o.0(Eq.  1875) 
a(Eq.  1910) 

11 

45  46-120 

11 

45 

58-080 

11 

47  35-810 

11 

47 

47-733 

a  +  JArt 

176°   40'-24 

176° 

43'-22 

cos  (a  +  |  Art) 
log# 
sec  J  Art 

log  AS 

AS 

9-999  26?* 
2-84612 
0 

9' 

2; 

999  29w 
846  12 
0 

2-845  3Sn 
-  700"-45 

2-845  41w 
-  700"-50 

S0(Eq.  1875) 
S(Eq.  1910) 

38° 
38 

36'  55"  55 
25  15  -10 

38 
38 

0  33' 
21 

33"-22 
52  -72 

The  next  step  is  to  make  the  corresponding  reduction  of  the 
proper  motion  from  the  one  equinox  to  the  other.  We  find 
from  (26) 

nat.  sin  £=+0000  25. 

nat.-cos£=      1 -000  00. 


144.] 


TRIGONOMETRIC  REDUCTION 


275 


The  two  computations  then 

are  : 
A. 

B. 

log/*a 
log  cos  30 

log  /xa  cos  <50 

2-71009 

9-89285 

2-70941 
9-89318 

2-60294 

2-60259 

/xa  cos  80 

400"-81 

400"-49 

/zs  sin  S 
pa'  cos  8 

-0  -14 

-0  -14 

400-67 

400-35 

log  „     „ 
log  sec  8 
log  fj>a' 

2-60279 
0-10598 

2-60244 
0-10564 

2-70877 

2*70808 

/*a'(Eq.  1910) 

oir-4i 

510"-60 

=  34-094  s. 

34-040  s. 

/zscos£  =  />is 

-577-97 

-578-19 

-  //a  cos  8  sin  $ 
/V(Eq.  1910) 

0-10 

-      0-10 

-578-07 

-578-29 

The  results  (B)  are  the  final  ones.  But  results  (A)  are  not 
because  they  give  the  position  and  motion  of  the  star  on  the 
sphere  at  the  initial  epoch,  referred,  however,  to  the  equinox  of 
1910.  We  therefore  continue  the  computation  (A)  by  finding 
the  change  in  oc,  S,  and  /x,  due  to  the  proper  motion  of  the  star 
on  the  sphere  during  the  interval  between  the  epochs. 

Beginning  with  the  proper  motions,  we  shall  find  the  centen- 
nial variations,  and  hence  the  reductions  to  be  almost  the  same 
as  in  the  first  computation.  Thus  we  have  for  the  centennial 
proper  motions  for  epoch  1875,  and  equinox  of  1910 : 

Computation  (A),      -          /xa'  =  34*094  s.     M6'  =  -  578"'07 
Change  in  35  years,  -         -     -   0*054  -     0  '22 

Motions  for  1910,      -  34-040  -578  '29 

Results  of  Computation  B         34-040  -578  -29 

The  two  results  should  in  theory  be  the  same. 
Next,  to  reduce  position  (B),  the  mean  of  the  proper  motions  (B) 
for  the  two  epochs  are  found  to  be : 

Ma  =  34s-067;  M8=  -578"18. 

These,  multiplied  by  0*35,  give  for  the  reductions  from  1875 
to  1910,  Aoc=  + 11-923  s. ;  Ac$=  -3'  22//-36. 


276  MEAN   PLACES  OF  THE   FIXED   STARS  [§144. 

These,  applied  to  the  results  of  computation  (A),  namely : 

h.         m.  s. 

a=ll  47     35-810;  (5  =  38°    25'    15"'10, 

give              a=ll  47     47*733;  £  =  38     21     52-74. 

(B)  gave                         47-733  52  -72. 

Discrepancy                       '000  -02. 

Reduction  of  ft  Ursae  Minoris  from  1755  to  1875. 
As  a  second  example,  we  reduce  the  position  of  ft  Ursae 
Minoris  for  1755,  as  found  in  Auwers'  catalogue  of  Bradley 
stars,  from  that  epoch  to  1875,  an  interval  of  120  y.  Omitting 
proper  motion,  we  give  the  computation  both  by  the  rigorous 
formulae  and  the  development  in  powers  of  Ap.  The  position 
for  1755  is :  a0=  14  h.  51  m.  42'56  s.  =  222°  55''64, 

($0=+7509'23"-2. 
The  constants  of  reduction  are : 

f0  =  46'-05,       m  =  6  m.  8'495  s., 
log  h  sin  0  =  2-205  27,    0  =  40'  6'H2. 
Whence  a  =  223°  41'-69. 


The  two  computations  are  : 

Rigorous. 
tan|0     7-766 
cos  d    y*ooy?i 

Approximate, 
log  tan  80        0-57672 
log  h  sin  9        2-20527 

tan  \Q  cos  a 
log  tan  80 
Diff. 
Subt.log(Tab.  B) 

sin0 

P 
sin  a 
p  sin  a 
cos  a 

1  —  p  cos  a 
tan  Aft 

7-625n 
0-576713 
2-952 
-486 

log  cos  a 
Arg. 
log#  (Tab.  I.) 
log  sin  a 
logA0a 

V 

\a  (Tab.  II.) 

/V(Tab-  III.) 
Red.  (Tab.  IV.) 
Aa 

2-78199 
9-859  167i 

2-641  15n 
-•013  61 
9-839  36n 

0-576227 
8-066  937 
8-643164 
9-839  36371 
8-482  527n 
9-85916/1 
8-502  32ri 
0-013592 
8-468935 

2-607  747i 

-  405-26 
4-      0-459 
0-028 
+      0-117 
-  404-712 

Aa-l°  41'  10"-64 


144.] 


TRIGONOMETRIC  REDUCTION 


277 


Rigorous. 

h. 

Aa=  -0 

m  +0 
oc0  14 


m.       s. 
6  44-709 
6     8-495 
51  42-56 


(Eq.  1875)0.  14   51     6-35 


Approximate. 

h. 

m. 

s. 

Aa  = 

-     0 

6 

44-712 

m 

+   0 

6 

8-495 

oc0 

14 

51 

42-56 

oc 

14 

51 

6-34 

W-°>} 

tan|0 
cos  |  (a'  +  a) 
sec  J(a'-a) 
tani(S-S0) 


Declination. 

9990   rti'.-iA 

££  £        ly  J.       i.  \J 

-0   50-59 

7-765921 
9-865  173ft 
47 


Declination. 
JAa     -  0°  50'-59 
a  +  AAft     222    51  -10 


log* 
cos  (a  +  |Aa) 
sec  JAo, 
log  AS 

AS 
S 

3 

9 

•381  372 
•865  173ft 
47 

3-246  592ft 

-  !764"-38 
-  0°  29'  24"-38 
75     9  23  -20 

74  39  58  -82 

7-631 141ft 
1(5  _so)  _o°  14'42a-19 
S  -  S0  -  0   29  24  -38 
S0    75      9  23  -20 
(Eq.  1875)  S   74   39  58  -82 


Reduction  of  Polaris  to  2100. 

As  a  case  where  the  original  form  (14)  of  the  equations  of 
reduction  is  most  convenient,  let  us  reduce  the  mean  place  of  the 
pole- star  to  2100,  an  epoch  two  years  before  the  nearest  approach 
of  the  pole  to  the  star.  The  position  and  centennial  proper 
motion  for  1900  are : 

oc  =  lh.  22m.  3319s.,        yua=  +13'64  s., 
S  =88°  46'  26"-61,  /xs=  +0"'33. 

We  first  make  the  reduction  for  proper  motion  during  the 
interval  of  200  years. 

AiO.  =  2-00  X  Ma  =  +  27-28  s., 
A1(S=2-OOx/x8=+0//-66. 

In  the  equation  (10)  the  last  terms  will  be  negligible.  We 
therefore  have,  for  the  position  of  (2100)  referred  to  the  origin 

Ofl900'  <x0  =  lh.23n,047S., 

<L  =  88°46'27"-27. 


278  MEAN   PLACES   OF  THE   FIXED  STARS 

The  constants  are  (App.  IV.) : 

f0  =  l°  16'  49"-8, 
z  =  l  16  53  -0, 
0  =  1  6  47  '3. 


[§  144. 


20°  45     7"-05 

1     16  49  -8 

22      1  56  -85 


sin  a  9-5741839 

cos50  8-3302491 

cos  a  9-9670664 

sin  6  8-288  399  4 

sinS0  9-9999006 

cos<9  9-9999180 


cos  0cosS0cosft 
sin  6  sin  8Q 
diff. 

8-2972335 
8-288  300  0 
0-008  933  5 

subt.  log 
cos  8  cos  a' 

1-6822900 
6-6060100 

cos  6  sin  a' 

7-9044330 

tana' 

1-2984230 

a'  87°     T  13"-40 
z     1     16    53  -0 
a(Eq.  2100)  88    24      6  -4 
=  5h.  53m.  36-43  s. 

sin  ft'       9-9994512 

cosS       7-9049818 

8(Eq.  2100)  89°  32'  22"'66 


It  may  be  inferred  from  these  results  that  the  pole  will 
the  star  early  in  2102  at  a  distance  of  27'  36"'7. 


Section  III.    Development  of  the  Coordinates  of  a  Star 
in  Powers  of  the  Time. 

145.  The  rigorous  methods  developed  in  the  two  preceding 
sections  are  necessary  when,  owing  to  the  great  length  of  the 
interval  through  which  the  reduction  extends,  or  to  the  high 
declination  of  the  star,  the  change  of  its  coordinates  is  an 
appreciable  fraction  of  its  polar  distance.  In  ordinary  cases 
the  method  of  developing  the  coordinates  of  the  star  in  a  series 
proceeding  according  to  the  powers  of  the  time  is  generally 
adopted. 

Our  first  problem  is  to  express  the  rate  of  change  of  the 
coordinates  in  terms  of  the  elements  of  position  and  motion  of 
the  star  and  of  the  pole.  Referring  to  the  equations  (16),  and 


§  146.]  DEVELOPMENT  OF  THE  COORDINATES  279 

treating  the  interval  of  time  and  the  motions  between  the  two 
epochs  as  infinitesimal,  we  see  from  §  138,  Eq.  (16)-(19),  that 

a  reduces  to  cc  +an  infinitesimal, 

0  reduces  to  ndt, 

p  becomes  ndt  tan  J, 
a  —  a  reduces  to  p  sin  a  =  ndt  sin  a  tan  S, 
£0  +  2  reduces  to  mdt.     (See  §  125,  Eq.  14.) 

We  therefore  have 

a' — a  =  (m  -f  n  sin  a  tan  8)dt. 
Also,  since  a' —  a  becomes  infinitesimal, 

J(a'~|-a)  becomes  oc  +an  infinitesimal, 
so  that  S'  —  8  =  ndt  cos  a. 

Adding  the  proper  motions,  the  differential  coefficients  of  the 
coordinates  as  to  the  time  become 

DtcjL  =  m  +  n sin  OL  tan  3  +  /ma  =  pa  +  fjLa\  /oo\ 

DtS  =7lCOSa  +  /Xs=£>8  +  /X5  J 

which  will  be  the  coefficients  of  t  in  the  development. 

146.  The  secular  variations. 

To  form  the  coefficients  of  the  second  power  of  the  time,  we 
have  to  differentiate  these  last  expressions  as  to  the  time. 
Taking  first  the  precessions  pa  and  ps,  in  (28),  we  find 

Dtpa  =  Dtm  +  sin  OL  tan  3  Dtn  +  n(pa + /xa)  cos  a  tan  S\ 

+  n(ps  +  iu.s)smoL  setfS  L..(29) 

Dtp8  =  cos  a.  Dtn  —  (  pa + /xa)  n  sin  a  J 

The  corresponding  changes  in  yua  and  JULS  comprise  two  parts : 
one  due  to  the  proper  motion  of  the  star,  found  in  §137,  the 
other  to  precession.  The  combined  effect  of  the  two  motions 
upon  the  proper  motion  itself  may  be  found  by  the  equations 
(8)  and  (27),  taking  S  in  the  latter  as  infinitesimal.  We  then 

have  sinS=S  =  ndtsma.sQc8,  (30) 

cos$=l, 

yU/COS  S'  —  yUaCOS  S  = 


280  MEAN  PLACES  OF  THE   FIXED   STARS  [§  146. 

or,  since  the  first  member  of  this  equation  is  the  infinitesimal 
increment  of  /xa  cos  S, 

cos  Sd/uia  —  yua  sin  SdS  =  ^ii  dt  sin  oc  sec  S. 
In  cZ<5  we  are  to  include  only  the  precession  p&.     Hence 

cos  8  Dt/uLa  =  iuLan  cos  oc  sin  S+jmsn  sin  oc  sec  S  .............  (31) 

Dividing  by  cos  8,  we  have  the  required  variation. 
We  have,  in  like  manner,  from  (27)  and  (30),  for  the  infini- 
tesimal increment  of  yu§, 

PS  —  MS  =  dps  =  —  fjLan  dt  sin  a, 
whence  Dt^  =  —  /uLan  sin  oc  ............................  (32) 

The  variations  (31)  and  (32)  are  those  due  to  the  precession 
alone.  Adding  them  to  the  corresponding  values  (8),  which 
give  the  variations  due  to  the  proper  motions  alone,  we  find  for 
the  entire  variations  of  the  proper  motions, 

n  8     } 

(33> 
=  —  Van  sin  oc  —  /*a2  sin  S  cos  S 


The  sum  of  the  first  equations  of  (29)  and  (33)  gives  the 
second  derivative  of  the  R.A.: 


.(34) 


-172-  =  Dtm + Dtn  sin  oc  tan  S 

+  n(pa -f 2/xa)cos  oc  tan  S 
+ n  (p&  +  2yu8)  sin  a  sec2  £ 
+  2fjiafjLS  tan  (5 

For  the  declination,  we  find  in  the  same  way, 

dzS 

-T72  =  Dtn  cos  a  —  n(pa + 2/za)sin  a  —  \ ^ia2sin 2$ (35) 

In  practical  application  we  reduce  the  constants  which  enter 
into  these  expressions  to  numbers.  It  will  also  be  found  con- 
venient in  the  terms  of  each  expression  which  contains  Dtn  ix> 
make  the  substitution 


sin  oc  tan  3  =  — , 

n 

Ps 

cos  oc  =  — . 
n 


§  146.]  THE  SECULAR  VARIATIONS  281 

These  terms  then  become, 

.    -r,  .          ?7i  n  Dt 

mRA.;   --Dt 


in  Dec.  ;  —  Dtn. 


n 


In  astronomical  literature  it  is  common  to  express  ra  and  n 
with  reference  to  the  year  as  the  unit.  But  for  the  longer 
intervals  over  which  the  reductions  of  stars  must  hereafter 
extend,  it  will  be  more  convenient  to  adopt  the  solar  century  as 
the  unit.  This  is  especially  desirable  in  the  case  of  the  proper 
motions,  because,  in  the  great  majority  of  cases,  the  annual 
motions  are  so  small  as  to  require  an  inconvenient  number  of 
O's  after  the  decimal  in  their  expression.  We  shall  therefore 
compute  the  numerical  coefficients  in  terms  of  the  century  as  the 
unit,  and,  to  avoid  any  possible  confusion,  write  the  centennial 
values  of  m  and  n  as 

mc=100m, 


thus  retaining  m  and  n  as  the  annual  values. 

We  take  1900  as  the  epoch  for  which  the  coefficients  are  to  be 
given.     For  this  epoch  we  have,  from  Appendix  III., 


dmc 


=  4608"-50   =  307-234  s., 
=  2004  -68   =133-646, 

=    +2-79   =+0186, 


dt 

=    -0-853  =-0-057. 


In  the  expressions  of  the  required  derivatives  as  above 
written,  the  quantities  are  supposed  to  be  expressed  in  homo- 
geneous units,  say  seconds  of  arc.  But  in  astronomical  practice 
all  the  terms  relative  to  the  R.A.  are  expressed  in  seconds  of 
time.  We  must  therefore  multiply  or  divide  the  coefficients  by 
15  in  such  a  way  that,  in  the  second  members,  pa  and  ,aa  shall  be 
expressed  in  time  and  ps  and  /ms  in  arc,  while  the  results  shall 
give  the  derivatives  of  a  in  time  and  those  of  S  in  arc. 


282  MEAN   PLACES   OF  THE   FIXED  STAKS  [§  146. 

Thus,  assuming  that  pa  and  /xa  are  expressed  in  time,  we  first 
multiply  them  by  15  to  reduce  them  to  arc.  Then,  all  the 
terms  will  give  the  values  of  the  tirst  members  in  seconds  of  arc. 

Then,  to  reduce  the  second  derivative  of  a.  to  time,  we  divide 
all  its  terms  by  15.  In  the  last  term  of  each  expression,  which 
is  of  two  dimensions  in  /x,  we  must,  to  produce  homogeneity, 
multiply  the  coefficients  by  sin  1".  In  this  way  the  reduction 
to  numbers  gives  us 


~  =  0-317  s.- 


+  [7'9876](pa+2Ma)cos  octant 


.(36) 


+  [4-9866]  yua,us  tan  S 

g|=-  [6-6289]^ 

-[9-1637](2>a+2Ma)sina 
-[6-7367]Ma2sin2<$ 

In  these  expressions  all  the  logarithms  are  to  be  diminished 
by  10.  It  may  be  noted  that  the  last  term  in  these  expressions 
can  scarcely  be  sensible  except  in  the  extreme  case,  when  /j.  tan  S 
amounts  to  several  hundred  seconds  of  arc. 

If  a  great  number  of  these  quantities  are  to  be  computed, 
the  work  may  be  facilitated  by  tabulating  the  quantities  : 


=  [7-9876]  tan  S 


1 


=  0-317  -[6'6289]pa 


Cs=  -[6-6289]p8  =  -[9-9310]coso 
Tables  for  A,  5,  Ca,  and  C$  are  found  in  Appendix  III. 

147.  It  is  the  common  practice  in  catalogues  of  stars  'to  give 
the  annual  variations,  or  precessions,  and  the  secular  variations 
•of  these  quantities,  that  is,  their  rate  of  change  per  century. 
These  secular  variations  are  equal  to  the  preceding  ones  divided 
by  100. 


§  148.]  THE  SECULAR  VARIATIONS  283 

When  the  century  is  taken  as  the  unit,  as  in  the  case  of  the 
preceding  development,  the  expression  for  the  first  two  terms  of 
the  reduction  to  an  epoch  T  is : 

dt* 


When  annual  variations  and  secular  variations  are  given,  the 
formis:  ./da.     It  \ 


Iii  either  case  the  result  may  be  obtained  by  multiplying  the 
variation  for  the  mid-epoch  by  the  elapsed  time. 

148.  The  third  term  of  the  reduction. 

To  obtain  the  coefficients  of  t3  in  the  expression  of  the  co- 
ordinates, we  have  to  differentiate  the  expression  for  the  second 
derivative.  If  the  effect  of  proper  motion  is  included,  the 
formulae  thus  derived  are  too  prolix  for  practical  use. 

It  may  be  doubted  whether,  in  ordinary  practice,  the  actual 
computation  of  this  term  is  the  best  course  to  follow.  The  writer 
has  always  found  the  easiest  course  to  be  to  compute  a  second 
value  of  the  second  term  for  the  terminal  epoch,  using  an 
approximate  position  of  the  star,  and  then  making  use  of  the 
following  simple  form,  easily  derived  by  taking  the  half  sum  of 
the  two  Taylor's  series  formed  by  interchanging  the  epochs, 


_       ^  !    t*d3QL 

dt      2    dt2  +6  dt*' 

dSOL 


In  the  case  of  an  isolated  star,  it  may  be  yet  easier  to  compute 
the  precession  for  three  equidistant  epochs,  those  of  reduction, 


284 


MEAN   PLACES   OF  THE   FIXED   STAES 


[ 


and  the  mid-epoch.    Calling  the  precessions,  in  the  order  of  time, 
p0,  p1  and  p2,  the  expression 


will  then  give  the  total  reduction.  This  method  was  adopted  by 
Auwers  in  reducing  the  positions  of  the  Bradley  stars  from  1755 
to  1865. 

If,  for  any  reason,  this  avoidance  of  the  third  term  is  either 
inaccurate  or  troublesome,  we  may  always  have  recourse  to  the 
rigorous  trigonometric  reduction. 

As  an  example  of  the  computation  and  reductions,  let  us  take 
the  position  and  proper  motion  of  1830  Groombridge  for  1875,  as 
given  in  §  144,  and  develop  the  R.A.  in  powers  of  the  time  to  T2. 
We  find,  from  these  data,  the  precessional  motions  in  Appendix  III., 
and  the  formulae  and  tables  for  the  secular  variations,  the 
motions  for  1875  and  1910,  as  in  the  following  computation  : 


1875-0. 

K.A.,  a  =  176026'31"-8 
Dec.,  8=   38   36  55  -6 


smoc 
tan  8 


nc  sm  OL  tan  S 
me 

Pa 
Pa 


8-792784 
9-902  401 
2-126001 
0-821 186 

6-625  s. 

307-187 

313-812 

34-198 

348-010 


cosoc       9-999162, 


382-21  s. 
ps    -2001"-03 
^   -•    577  -97 
AS    -  2579  -00 
2/z*    -3156  -97 


1910-0. 

176°  56'  56"-2 
38    21   52  -8 

8-726  122 
9-898498 
2-125941 
0-750561 

5-631  s. 

307-252 

312-883 

34-040 

346923 

9-999  384n 
3-301  412W 

380-96  s. 

-  2001"-76 

-  578  -29 

-  2580  -05 

-  3158  -34 


149.]      PRECESSION   IN   LONGITUDE   AND   LATITUDE 


285 


1875-0. 

-  2/xa)       2-5823 
log  cos  a       9-9992, 
log  A  (Tab.)       7-8900 
log  (2)       0-4715, 


3-4993, 
log  sin  a      8'7928 
logj5(Tab.)       7-0258 
log  (3)       9-3179n 


log/Xtt 
log/Otj 

tan  8 

log  coeff. 
log  (4) 

(1) 
(2) 
(3) 
(4) 

1-534 
2-762 
9-902 

4-198 
4-987 

9-185 

+  0-183 
-2-961 
-  0-208 
-0-153 

-3-139 


1910-0. 

2-5809 

9-9994n 

7-8861 

Q-4664n 

3-4995H 
8-7261 
7-0228 
9-2484n 

1-534 

2-762 
9-898 
4  194 


9-181 

+  0-184 
-2-927 
-0-177 
-0-152 
-3-072 


149.  Precession  in  longitude  and  latitude. 

We  now  investigate  the  instantaneous  rate  of  change  in  the 
longitude  and  latitude  of  a  star  due  to  the  precessional  motion. 


FIG.  34. 


Let  NftQ  be  the  position  of  the  ecliptic  at  the  initial  moment ; 
the   position   at  the   moment    following;   EQ,   E  the   two 


286  MEAN   PLACES   OF  THE    FIXED   STARS  [§149. 

equinoxes;  SR0,  SR  perpendiculars  dropped  from  the  position 
of  the  star  S  upon  the  ecliptics  at  the  two  moments  ; 

X0,  X,  the  longitudes!    ,   ,  , 

°      '   _     .     .°    .         I  at  the  two  moments. 
Po,  p,  the  latitudes    ) 

We  then  have         \0  =  E0R0  =  NR0-  NE0, 
\ 


and  for  the  increment  of  X, 

^\  =  NR-NR0+NE0-NE  ................  (a) 

In  the   infinitesimal   triangles   SRR0  and   NRR^,   using  the 
notation  of  §  122,  we  have 

K  =  angle  R0NR, 

I  =  NE0  —  NE  =  general  precession, 


A  study  of  the  relations  of  these  triangles,  putting  S  for  the 
infinitesimal  angle  at  S,  and  applying  Theorem  ii.  of  §  7,  leads 
to  the  equation  NS-N^-Sianp,  ........................  (6) 


while,  noting  that  R  and  R0  are  right  angles,  Theorem  iii.  gives 

the  equation         S  cos  ft  =  K  cos  NR  =  K  cos  (X  +  A7C), 

whence  $  =  Ksec/3cos(X  +  A70)  ......................  (c) 

A  comparison  of  (a),  (6),  and  (c)  gives 

AX=  -/ctan/3cos(X  +  ^0)  +  £  .............  (39) 

From  the  triangle  RSR0,  we  have 

A/3  =  SR-SR()  =  Ksin(\  +  N0).    .  ...........  (40) 

We  may  treat  the  quantities  AX  and  A/3,  K  ,  and  I  as  derivatives 
with  respect  to  the  time,  in  accordance  with  the  methods  of 
Chapter  IX.  For  the  epoch  1850  we  have,  from  (4)  of  §  123, 


§  149.]      PRECESSION   IN   LONGITUDE   AND   LATITUDE  287 

Interpolating   to   1900,  the  epoch   now  most  generally  used, 
we  have  ^0  =  6°  44\S1     (Eq.  1850) 

Free,  for  50  years,  -0   41  '87 

N0  for  1900,  6°    2'«94 

We  find  from  the  same  data, 

AC  for  1900  =  47"-107. 

The  complete  expression  for  the  rate  of  change  at  1900  due  to 
precession  alone  now  becomes 

for  1900, 

(41) 


the  unit  of  time  being  the  solar  century. 

These  equations  give  only  the  rate  of  change  for  1900,  or  the 
coefficients  of  the  first  power  of  T.  To  find  the  coificients  of  the 
second  power  of  the  time  we  must  differentiate  the  expressions 
(41)  as  to  the  time. 

The  angle  A-f  jV0  is  the  distance  from  the  instantaneous  node 
of  the  moving  on  the  fixed  ecliptic  to  the  projection  of  the  star 
on  the  ecliptic.  It  therefore  changes  only  in  consequence  of  the 
motion  of  this  node  and  the  proper  motion  of  the  star.  The 
former  is,  from  (4)  of  §  123, 


and  the  centennial  motion  of  A  +  JV0  is 

m  +  29'-0  =  f^"  sin  1"  -f  0-008  43, 
yux  being  the  proper  motion  in  longitude. 

flic 

We  also  have  |^=  -  0"'068, 

and,  by  differentiation  of  (40)  and  substitution, 

r/2/Q 

^  =  0"-40  cos  (A  +  N0)  -  0"-07  sin  (A  +  N0) 

=  0"-40cos(A  +  16°-2), 
the  effect  of  proper  motion  being  neglected. 


288  MEAN   PLACES   OF  THE   FIXED   STAES  [§  149. 

In  a  similar  way  we  find,  for  the  part  of  the  second  derivative 
of  X  arising  from  the  variation  of  X  and  JV0, 

0"40tan/3eos(X  +  16°-2). 

The  motion  of  ft  gives  rise  to  a  term  having  the  coefficient 
0"'005.  As  there  is  rarely  any  occasion  for  using  the  ecliptic 
coordinates  of  stars  outside  the  limits  of  the  zodiac,  this  term 
may  be  dropped. 

NOTES  AND   REFERENCES. 

Two  sets  of  tables  ha\7e  recently  been  published  for  the  rapid  computation 
of  the  annual  precessions  of  the  stars,  the  secular  variations  and  the  co- 
efficients of  the  third  power  of  the  time.  They  are  : — 

DOWNING,  A.  M.  W.,  Precessional  Tables  adapted  to  Newcomb's  value  of  the 
Precessional  Constant  and  Reduced  to  the  Epoch  1910.  Edinburgh,  Neill  &  Co., 
1899. 

BECKER,  Tafeln  zur  Berechnung  der  Praecession  (Extract  from  the  Annals 
of  the  Strassburg  Observatory,  volume  ii.).  Karlsruhe,  G.  Broun,  1898. 

Becker's  tables  are  based  on  the  Struve-Peters  values  of  the  precessional 
motions  for  the  fundamental  epoch  1900.  In  connection  with  the  tables  is 
given  the  reduction  to  the  new  adopted  values  of  the  precessional  motions. 
The  secular  variations  in  the  two  theories  differ  but  slightly.  The  third 
term  in  the  precession,  that  containing  the  factor  1 3,  is  scarcely  sensible  for 
declinations  less  than  40°,  unless  the  reduction  extends  over  more  than  one- 
half  a  century.  The  writer  conceives  that,  where  account  has  to  be  taken  of 
this  term,  it  will  be  easier  to  compute  the  trigonometric  reduction  by  the 
tables  of  the  present  work  than  to  use  the  third  term.  But  if  it  is  desired 
to  use  this  term  tables  of  the  coefficient  of  T3  itself  with  the  double  argument 
K.A.  and  Dec.  will  be  found  in  some  of  the  Introductions  to  the  star 
catalogues  of  the  Astronomische  Gesellschaft,  quoted  in  chapter  xiii. 

The  fundamental  catalogue  in  Astronomical  Papers  of  the  American 
Ephemeris,  vol.  viii.,  gives  tables  for  the  trigonometric  reduction  to  six 
places  of  decimals,  but  they  do  not  extend  to  so  high  a  declination  as  those  in 
Appendix  of  the  present  work. 


CHAPTER   XL 
REDUCTION  TO  APPARENT  PLACE. 

Section  I.    Reduction  to  Terms  of  the  First  Order. 

150.  Reduction  for  nutation. 

The  theory  of  nutation,  or  of  the  revolution  of  the  true 
celestial  pole  around  the  mean  pole,  has  been  developed  in 
Chapter  IX.  We  have  now  to  determine  the  effect  of  this 
motion  upon  the  R.A.  and  Dec.  of  a  heavenly  body.  The 
relation  of  the  true  to  the  mean  pole  is  expressed  by  the  two 

quantities : 

A^/r,  nutation  in  longitude. 

Ae,  the  nutation  of  the  obliquity  of  the  ecliptic. 
The  principal  terms  of  the  original  expressions  from  which 
A\^  and  Ae  are  derived  have  been  given  in  Chapter  IX.,  §  134, 
and  are  more  completely  tabulated  in  the  Appendix.  The  funda- 
mental data  derived  from  them  may  be  found  in  the  annual 
Ephemerides. 

Since  the  nutation  does  not  affect  the  position  of  the  ecliptic 
itself,  the  latitude  of  a  heavenly  body  is  not  affected  by  it. 
For  the  same  reason  the  foot  of  the  perpendicular  from  the 
body  to  the  ecliptic,  and  therefore  the  position  of  this  foot, 
remains  unchanged.  Hence  the  effect  upon  the  longitude  of  a 
body  is  only  to  increase  it  by  the  quantity  A^. 

151.  Nutation  in  R.A.  and  Dec. 

The  effect  of  nutation  upon  the  R.A.  and  Dec.  when  the  star 
is  not  very  near  the  pole  is  so  small  that  its  powers  may  be 
N.S.A.  *  T 


290  EEDUCTION  TO  APPARENT  PLACE  [§  151. 

neglected.  It  is  then  at  once  obtained  from  the  formulae  of 
§  54  by  substituting  in  the  equations  (27)  for  d\  and  de  the 
values  of  Ai/*-  and  Ae  produced  by  the  nutation.  The  following 
are  the  terms  of  (27)  which  thus  come  into  use  : 

cos  8  Aoc  =  cos  S  cos  /3A  A  —  sin  S  cos  ocAel  -,  ^ 

A<S  =  sin  8  cos  /3AX  +  sin  ocAe         J  ' 
From  the  parts  of  the  triangle  EPS  in  §  51,  we  have 

sin  S  cos  /3  =  cos  oc  sin  e  ^  ,~\ 

cos  S  cos  /3  =  coe  e  cos  S  +  sin  e  sin  S  sin  ocj 

Substituting  (2)  in  (1)  and  putting  A\/r  for  AX,  we  find,  for 
the  nutation  of  the  R.A.  and  Dec., 

Aoc  =  (cos  e  +  sin  e  sin  a  tan  S)  Ai/r  —  cos  oc  tan  £Ae)  ^\ 

A<5  =  cos  OL  sin  eA^  +  sin  aAe  / 

In  practice  the  reduction  for  nutation  is,  in  the  case  of  the 
fixed  stars,  combined  with  the  effect  of  precession  from  the 
beginning  of  the  solar  year.  As  already  mentioned,  it  is 
the  universal  astronomical  practice  to  refer  the  mean  places  of 
the  fixed  stars  to  the  equinox  and  equator  of  the  beginning  of 
some  such  year.  Then,  instead  of  dividing  the  reduction  for 
precession  and  nutation  into  the  two  parts, 

Precession  to  date  +  nutation, 
they  are  divided  into 

Precession  to  beginning  of  solar  year 

+  (Precession  from  beginning  of  year  to  date  +  nutation). 
The  two  reductions  in  parentheses  are  combined  into  one  in 
the  following  way  : 

Putting  r  for  the  elapsed  fraction  of  the  solar  year,  the 
changes  in  the  coordinates  of  the  star  due  to  precession  from 
the  beginning  of  the  year  through  the  time  T  are,  neglecting 
the  secular  variation, 


where  m  and  n  have  the  following  values  (§  125) 

m=_pcose  —  A'|  „. 

n=psiue        ) 

p  being  the  annual  rate  of  luni-solar  precession. 


§  151.]  NUTATION   IN  RA.   AND   DEC.  291 

The  corrections  (4)  with  the  substitution  of  (5)  are  now  to 
be  combined  with  (3),  the  nutation.  Putting,  for  the  moment, 
F  for  the  coefficient  of  A\/r  in  the  first  equation  (3), 

F=  cos  e  +  sin  e  sin  oc  tan  S. 
It  will  be  seen  from  (4)  and  (5)  that 

m  -f  n  sin  a  tan  S  =  pF—  X', 
whence  ^m  +  ^sinatan^x; 

The  sum  of  the  terms  of  Aoc  in  (4)  and  the  first  equation  of 
(3)  gives  for  the  total  change  in  R.A.,  due  to  the  combined  effect 
of  nutation  and  precession  from  the  beginning  of  the  year, 


(6) 

P 


So,  if  we  put          A=T-\ — — 

a = m + n  sin  oc  tan  3 


(7) 


P 

we  shall  have  the  effects  of  precession  from  the  beginning  of 
the  year  and  nutation  in  longitude  combined  in  the  simple 
expression  Aa  =  .4a+£.  ..............................  (8) 

For  the  declination  the  values  of  A£  in  (3)  and  (4)  may  be 
combined  in  a  similar  way.     We  have  from  (5), 

n 

sin  e  cos  oc  =  —  cos  oc, 

P 

and  thus  the  sum  of  the  two  terms  in  question  may  be  written 

;  ........  (9) 


So,  if  we  put  of  =  n  cos  oc,  ..............................  (10) 

we  shall  have  the  effects  of  precession  from  the  beginning  of 
the  year  and  nutation  in  declination   combined  in  the  simple 

form  M  =  Aa',.  ...(11) 


292  REDUCTION   TO  APPARENT  PLACE  [§  151. 

The  effect  of  nutation  in  obliquity  may  be  expressed  in  the 
same  way.     The  practice  is  to  put 


b  =  cos  a  tan  S  (or  numerically  y1-  cos  a  tan  S)  I (12) 

bf  =  —  sin  oc 

The  coefficient  6  is  divided  by  15  in  order  that  Aoc  may  be 
expressed  in  time. 

We  then  have,  for  the  nutation  in  RA.  and  Dec.  depending 
on  Ae, 

%°w} (13) 

152.  Reduction  for  aberration. 

The  formulae  for  the  effect  of  aberration  upon  the  coordinates 
of  a  fixed  star,  considered  as  infinitesimal,  are  found  in 
Chapter  VII.,  Eq.  (13).  We  note  that  the  terms  of  cos<SAa 
containing  e  as  a  factor  are  functions  of  a,  TT,  and  e,  and  being 
nearly  constant  in  the  case  of  any  one  star  are  regarded  as 
included  in  the  mean  R.A.  of  the  star,  and  left  out  of  considera- 
tion. We  thus  have  for  the  aberration : 

Aoc  =  —  K  sin  0  sin  oc  sec  8  —  K  cos  0  cos  e  cos  oc  sec  S\ 

A<5  =  —  K  cos  0  (sin  e  cos  S  —  cos  e  sin  oc  sin  8)  > (14) 

—  K  sin  O  cos  oc  sin  8  J 

If  we  put 

C=  —K  cose  cos  0  ] 

Z>=-/csin© 

c  =  cos  ocsec  8  -f- 15  (to  reduce  to  time)   |  Q  R\ 

d  =  sin  ocsec  8  +  15  (to  reduce  to  time)   j 
c'  =  tan  e  cos  8  —  sin  ocsin  8 


these  equations  become 


which  is  the  simplest  form  of  expressing  the  aberration  when  its 
powers  are  dropped. 


§  153.]  REDUCTION   FOR  PARALLAX  293 

153.  Reduction  for  parallax. 

When  we  take  into  account  the  effect  of  the  annual  parallax 
of  a  star  upon  its  R.A.  and  Dec.,  we  must  conceive  its  mean  place 
to  be  referred  to  the  sun,  and  then  find  the  reduction  to  the 
earth.  If  r  be  the  distance  of  the  star  from  the  sun,  and  X,  Y,  Z 
the  rectangular  equatorial  coordinates  of  the  sun  ;  and  if  we 
designate  the  geocentric  coordinates  of  the  star  by  accents,  they 
will  be  given  by  the  equations 

i    x'  =  r'  cos  S'  cos  OL  =  r  cos  S  cos  a.  +  X} 

2/'  =  r'  cos  $'  sin  oc'  =  r  cos  S  since  -f  Y\  ..........  P-*) 

z  =  r  sin  $'  =  r  sin  8  +  Z 

Owing  to  the  vast  distance  of  the  stars  and  the  consequent 
great  value  of  r,  we  may  treat  X,  Y,  and  Z  as  infinitesimal 
increments  of  x',  y',  and  z'  respectively,  and  determine  the  corre- 
sponding increments  of  oc  and  S  by  the  equations  (4)  of  §  48, 
putting  oc  and  S  for  X  and  6,  and  X,  Y}  Z  for  dx,  dy,  and  dz 
respectively.  We  also  put  TT,  the  annual  parallax:  of  the  star, 
that  is,  the  angle  subtended  by  'the  earth's  mean  distance  from 
the  sun  when  seen  from  the  star,  which  makes 


We  thus  derive,  from  the  equations  (4a)  of  §  48, 

cos<5Aoc  =  sin7r(  —  Xsinoc+  Fcosoc) 

A<5  =  sin  7r(Z  cos  S  —  X  sin  S  cos  a  —  Fsin  S  sin  oc)  J 

These  expressions  may  be  reduced  to  the  form  of  the  other 
star  corrections  in  the  following  way.  Putting,  as  before,  0  for 
the  sun's  true  longitude  and  R  for  its  radius  vector,  we  have 

X  =  RcosQ, 
Y=  jRcosesin  O, 
Z=R  sine  sin  0. 

Substituting  these  values  in  (18)  and  putting  TT  for  its  sine, 
we  find 

Aa  =  ^7r(  —  cos  O  sin  a  +  cose  sin  ©  cosoc)sec£ 
A(5=  RTT(  —  cos  O  sin  8  cos  oc  —  cos  e  sin  0  sin  $  sin  a. 
+  sine  sin  0  cos<5) 


294  REDUCTION  TO  APPARENT  PLACE  [§  153. 

These  can  be  expressed  by  means  of  the  same  star  constants 
as  are  used  in  computing  the  aberration,  after  multiplying  them 
by  the  parallax.  That  is,  if  we  put,  as  functions  of  the  coordi- 
nates of  the  star  and  of  its  parallax,  using  IT"  as  the  parallax  in 
seconds  of  arc  and  7rs  =  7r'/-7-15  =  '7r  in  seconds  of  time, 

Cj  =  ?rs  cos  oc  sec  S  —  TT"C 

dl  =  TTS  sin  a  sec  8  =  Tr"d 

c\  =  TT"  (tan  e  cos  S  —  sin  oc  sin  S)  = 

d\  =  TT"  cos  a  sin  S  =  Tr"d' 

and,  as  factors  depending  on  the  sun's  longitude, 

C1  =  R  cos  e  sin  0 1 
D^-^cosO      y 
we  shall  have 

A  _.        n  .     i    T%   .7 

(22) 


154.  Combination  of  the  reductions. 

We  next  show  how  the  preceding  reductions  may  best  be 
combined.  Omitting  the  reduction  for  parallax,  which  need  be 
taken  account  of  only  in  a  few  exceptional  cases,  the  reduction 
of  a  star  from  its  mean  place  at  the  beginning  of  a  year  to  its 
apparent  place  at  any  time  during  the  year  may  be  computed 
by  the  formulae  (8),  (11),  (13),  and  (16).  Adding  the  correction 
for  proper  motion  from  the  beginning  of  the  year  to  the  date, 
we  shall  have 


The  coefficients  A,  B,  (7,  D,  and  E  are  functions  of  the  time 
but  independent  of  the  position  of  the  star.  Hence,  on  any  one 
date,  they  are  the  same  for  all  the  stars.  They  are  known  in 
astronomy  as  the  Besselian  day  numbers,  after  the  great  Bessel, 
who  first  introduced  them  into  use.  Their  values  for  every 
day  of  the  year  are  found  in  the  annual  ephemeris. 

On  the  other  hand,  the  numbers  a,  a,  6,  etc.,  being  functions 
of  the  place  of  the  star,  are  regarded  as  constants  for  greater 
or  less  periods  of  time.  The  logarithms  of  these  constants  for 


§  155.]  COMBINATION  OF  THE   REDUCTIONS  295 

individual  stars  are  given  in  some  of  the  catalogues,  so  as  to 
save  the  astronomer  using  the  catalogue  the  trouble  of  com- 
puting them.  But  as  the  position  of  every  star  varies  from 
year  to  year,  it  is  a  question  how  long  any  such  constants  can 
be  used  without  important  error.  The  general  rule  is  that, 
in  the  case  of  stars  near  the  equator,  say  those  whose  declination 
is  less  than  45°,  the  constants  may  be  used  for  several  years 
unchanged.  But  as  we  approach  the  pole,  the  period  during 
which  no  change  need  be  made  becomes  shorter  and  shorter. 

Some  of  the  catalogues  give  in  addition  to  the  constants  for 
a,  given  epoch  either  their  values  at  some  other  epoch  or  the 
annual  change  in  the  last  figure  of  the  logarithm.  With  such 
catalogues  reductions  can  be  made  without  danger  of  error. 

155.  Independent  day  numbers. 

There  is  another  form  of  reduction  to  apparent  place  which 
is  much  used  when  sufficiently  accurate  values  of  the  star 
•constants  are  not  at  hand.  In  the  equations  (8),  (11),  and  (13) 
let  us  substitute  for  a,  a,  6,  and  b'  their  values  as  given  in  (7), 
{10),  and  (12).  The  reduction  for  precession  and  nutation  thus 
becomes  Aa  =  Am  +  (An  gin  a  +  g  cog  a^tan  ^  +  E, 

&S  —  AncosoL  —  BsiucL  j 

In  the  same  way,  the  terms  of  aberration  as  found  in  (14)  and 
(16)  may  be  written 

Ao.  =  ((7cosa+JDsina)sec<S  1 

A(5  =  C  tan  e  cos  8  +  (D  cos  oc  -  C  sin  a)sin  <?J 
In  the  second  term  of  (24)  let  us  replace  A  and  B  by  the 
quantities  g  and  G,  determined  by  the  equations 


g  cos  G  —  An)  ' 
we  shall  then  have 

A  n  sin  a  +  B  cos  oc  =  g  sin  (  G  +  OL), 


-and  (24)  becomes 

Aoc  =  g  sin  (G  +  oc)  tan  S  +  Am  +  E, 


296  KEDUCTION  TO  APPARENT  PLACE  [§  155. 

Let  us  also  transform  (25)  in  a  similar  way,  determining  h 
and  H  by  the  conditions 

hsmH=C}  ,2>i\ 

hcosH—Dj 
We  then  have 

C  cos  oc  +  D  sin  oc.  =  h  sin  (  H  -f  oc), 
D  cos  oc  —  G  sin  oc  =  h  cos(H+  oc), 
and  (26)  becomes 

Act  =  A  sin  (H+oC)  sec  (5, 
A<5  =  h  cos  (H  +  oc)  sin  <?  +  (7  tan  e  cos  & 
Let  us  also  put 

f=^  +  E\  ............................  (28) 

^  =  (7tane    j 

By  these  substitutions  the  total  reductions  for  nutation  and 
aberration,  adding  in  the  proper  motion,  become 


A£=#  cos(£+oc)+/t  cos(jy+oc)sin  8+i  cos  8  +  HJLST) 

which  may  be  used  instead  of  (23).    The  numbers  /,  g,  etc.,  known 
as  independent  day  numbers,  are  given  in  the  Ephemerides. 

The  choice  between  the  use  of  Besselian  and  of  the  inde- 
pendent day  numbers  depends  upon  the  special  character  of 
the  work.  The  general  rule  is  that,  if  the  problem  is  to  compute 
a  number  of  positions  of  the  same  star,  say  an  ephemeris  for 
an  entire  year,  the  Besselian  numbers  will  be  the  most  con- 
venient. This  advantage  will  hold  true  even  for  a  single 
apparent  place,  if  the  star  constants  a,  6,  etc.,  are  already  at 
hand.  But  if  these  constants  have  to  be  computed,  and 
especially  if  the  problem  is  to  reduce  a  large  number  of  stars 
to  apparent  place  at  the  same  date,  the  independent  day 
numbers  will  give  the  most  rapid  computation.* 


*The  computer  using  the  British  Nautical  Almanac  or  the  Connaissance  des 
Temps  should  have  in  mind  that  the  day  numbers  in  these  two  publications  have 
a  different  notation  from  that  above  used,  which  is  the  original  one  of  Bessel. 
When  these  numbers  were  introduced  into  England  by  Baily,  those  expressing 
aberration  were  changed  to  A  and  B,  and  those  for  nutation  to  C  and  D.  This 
system  was  also  adopted  in  Paris.  In  the  early  years  of  the  American  Ephemeris 


§  156.]   EIGOEOUS  REDUCTION  FOR  CLOSE  POLAR  STARS      297 

Section  II.    Rigorous  Reduction  for  Close  Polar  Stars. 

156.  In  the  preceding  method  of  reduction,  the  changes  pro- 
duced by  precession  during  the  fraction  of  the  year,  by  nutation 
and  by  aberration,  have  all  been  treated  as  infinitesimals.  It  has 
therefore  been  assumed  to  be  indifferent  whether  the  mean  or 
the  apparent  place  of  the  star  is  used  in  the  formulae,  and 
quantities  of  higher  dimensions  than  the  first  in  the  three 
changes  have  been  dropped  as  unimportant.  This  deviation 
from  rigour  will  lead  to  no  appreciable  error  when  the  amount  of 
the  reduction  is  not  an  important  fraction  of  the  star's  distance 
from  the  pole.  But,  however  small  the  changes  may  be  in 
themselves,  there  is  always  a  certain  distance  from  the  pole 
within  which  a  more  rigorous  process  is  necessary.  The  choice 
among  the  various  methods  of  reduction  that  may  be  adopted  in 
this  case  depends  largely  on  the  nature  of  the  problem  in  hand 
and  the  degree  of  precision  required. 

The  more  precise  methods  which  may  be  adopted  are  of  two 
classes.  In  one  a  formally  rigorous  reduction  is  carried  through 
by  trigonometric  methods.  In  the  other  class  the  reductions  are 
developed  to  quantities  of  the  second  order  with  respect  to  their 
values.  It  must  be  noted  in  this  connection  that  any  method  of 
development  in  powers  of  the  reduction  will  fail  in  the  immediate 
region  of  the  pole,  though  it  may  be  applicable  to  all  the 
standard  stars  now  in  use. 

In  order  to  appreciate  the  degree  of  precision  required,  the 
fact  must  be  borne  in  mind  that,  on  account  of  the  convergence 
of  the  meridians,  as  explained  in  §  44,  the  actual  error  in  the 
position  of  a  star  arising  from  a  given  error  of  its  R.A.  diminishes 
without  limit  as  the  pole  is  approached.  It  follows  that  if  we 
have  in  the  R.A.  an  expression  of  the  form 

Aoc  =  k  sec  S  or  Aoc  =  k  tan  S, 

the  English  system  was  adopted.  But  in  the  Berliner  Astronomisches  Jahrbuch, 
and  in  the  American  Ephemeris  after  the  first  few  years,  the  original  notation  has 
been  used  throughout,  as  defined  in  the  present  chapter.  It  may  also  be  said 
that  in  catalogues  in  which  polar  distance  is  used  instead  of  declination,  especially 
in  the  British  Association  catalogue,  the  accented  star  constants  for  the  declination 
have  their  sign  changed  in  order  to  give  the  reduction  of  the  polar  distance. 


298 


REDUCTION   TO   APPARENT  PLACE 


[§  156. 


then  although,  as  the  pole  is  approached,  Aoc  increases  without 
limit,  the  amount  of  correction  to  the  actual  position  of  the  star 
will  be  measured  by  k  only.  Since  it  is  impossible  in  practical 
measurement  to  gain  greatly  in  accuracy  by  being  near  the  pole, 
it  follows  that  the  importance  of  the  term  k  sec  3  must  depend 
on  the  value  of  k  alone. 

This  does  not  apply  to  a  correction  A$  in  declination.  If  this 
contains  a  factor  sec  3  or  tan  3,  it  will  increase  proportionally  to 
that  function.  Moreover  when  a  term  of  the  R.A.  contains  sec23 
or  tan2$,  the  effect  of  the  term  on  the  position  of  the  star 
increases  indefinitely  as  the  pole  is  approached. 

157.  Trigonometric  reduction  for  nutation. 

Let  P  be  the  mean  pole,  P'  the  actual  pole  as  affected  by 
nutation,  and  S  the  position  of  the  star.  It  is  indifferent  whether 


FIG.  35. 

we  take  for  P  the  mean  pole  of  the  date  or  that  at  the  beginning 
of  the  year.  It  will  be  generally  more  convenient  to  take  the 
pole  for  the  beginning  of  the  year.  Then,  as  heretofore,  the 
luni-solar  precession  to  date  will  be  combined  with  the  term 
of  the  nutation. 


§  157.]     TRIGONOMETRIC  REDUCTION  FOR  NUTATION  299 

Let  CP  and  O'P'  be  small  arcs  of  the  colures  through  P  and  P' 
and  PE  and  P'E'  arcs  of  the  circles  passing  through  the  mean 
and  apparent  equinoxes  respectively.  We  shall  then  have 

Angle  CPE= Angle  a'P'#'  =  90°, 


In  Fig.  35  the  day-numbers  g  and  G  are  geometrically  repre- 
sented, as  are  also  the  mean  and  reduced  coordinates  of  the  star, 
as  follows  : 

-G=RPP. 

-  G'  =  E'P'L,  P'L  being  the  continuation  of  PP. 


)  the  mean  R.A. 
a',  the  R.A.  affected  by  precession  to  date  and  nutation 


SQ  =  90°  —  PS,  the  mean  declination. 

<S'  =  90°  —  P'$,  the   declination   affected  by  precession    to 
date  and  nutation. 

From  Theorem  (ii.)  of  differential  spherical  astronomy,  we  have, 
assuming  that  P  is  the  pole  for  the  beginning  of  the  year,  and 
using  the  day  numbers  A  and  J5, 

PR  =  (pr  -f-  A^)  sin  e  =  Ap  sin  e  =  An. 

In  determining  g  and  G  from  p  +  A^  and  Ae,  we  may  treat  the 
triangle  RPP'  as  infinitesimal,  because  the  effect  of  the  resulting 
errors  will  be  only  an  error  of  the  second  order  in  the  position 
of  the  pole  P',  which  is  independent  of  the  position  of  the  star, 
and  therefore  does  not  increase  when  the  latter  is  near  the  pole. 

The  angle  G  and  the  side  PP'  =  g  may  therefore  be  found 
from  the  equations 

- 


#  cos  6r  =  Ap  sin  e  =  An 

From  Theorem  (iii.),  §  7,  we  have 

G'PL  =  CPP'  +  Ap  cos  e  =  GPP'  +f} 
the  term  E  in  /  being  dropped  because  unimportant  in  this  case. 


300  KEDUCTION  TO  APPARENT  PLACE  [§  157. 

Subducting  equal  right  angles,  we  shall  have  left 

G'=G-f. 
In  the  triangle  SPP'  we  have 

Angle  P  =  GCO+(T, 
Exterior  Angle  SP'L  =  a'  +  Q'. 

The  relations  between  the  five  parts  of  this  triangle  which  have 
been  defined  give  the  equations 

sin(90°  -  £>in(a'  +  £')  =  sin(900  -  50)sin(oCo  +  G), 
sin(900-(T)cos(a'-f  G')  =  cosg  sin(90°  -  <S0)cos(at)+  G) 

-sin#cos(900-(?0), 
cos(90°  -  cT)  =  sin  g  sin(90°  -  £0)cos(a0  +  G) 

+  cos0coB(900-<J0). 
Putting,  for  brevity,          a  =  oc0+  G, 


the  relations  become 

cos  <$'  sin  a'  =  cos  S0  sin  a,  ....  .............................  (30) 

cos  S'  cos  a'  =  cos  g  cos  80  cos  a  —  sin  g  sin  S  } 

"  > 

sin  8  =  sin  g  cos  ^0  cos  a  +  cos  g  sin  S0j 

These  equations  become  identical  in  form  with  (14),  §  138, 
when  we  write  g  for  9,  G  for  f0  and  G'  for  —0;  and  may 
therefore  be  solved  in  the  same  way.  But  g  is  so  minute,  its 
maximum  value  being  about  30",  that  we  may  drop  its  powers, 
when  not  multiplied  by  a  factor  which  becomes  infinite  at  the 
pole,  and  put  sin<7  =  <7,  cos<7=l.  With  this  change,  the  formulae 
for  solving  the  preceding  equations  for  OL  and  S'  are  as  follows. 
We  accent  the  symbol  p  to  avoid  confusing  it  with  the  precession 
and  put  An  for  the  increment  due  to  nutation  and  precession  : 


.(31) 


p  sin  a 
1/1  ~~1—  p'cosa 

Ana.  =  Ana  +  Ap  cos  e 


§158.]    TRIGONOMETRIC  REDUCTION  FOR  ABERRATION        301 
By  expressing  g  and  pf  in  seconds  of  arc,  computing 


we  may  use  the  Tables  of  Appendix  IV.  in  the  solution. 

It  is  also  to  be  noted  that  in  the  case  of  a  star  only  a  few 
minutes,  say  5'  or  less,  from  the  pole,  the  rigorous  equation  may 
be  necessary  in  the  computation  of  S. 

158.  Trigonometric  reduction  for  aberration. 

The  reduction  for  aberration  may  also  be  expressed  in  the 
trigonometric  form.  We  have  found  (§87)  that  the  changes  in 
the  equatorial  rectangular  coordinates  X^  Tv  Z1  of  a  star  pro- 
duced by  aberration  are  : 


(32) 
0 

R  being  the  distance  of  the  star  and  G  and  D  the  day  numbers. 
Expressing  the  spherical  coordinates  in  terms  of  the  rectangular 
ones,  putting  jR'  for  the  apparent  distance,  and 

R> 

J-R' 

we  find  that  the  apparent  R.A.  and  Dec.  a  and  8  may  be  derived 
from  oc'  and  3'  by  solving  the  equations 

/cos  S  cos  oc  =  cos  6"  cos  oc'  —  D} 

/  cos  £  sin  a  =  cos  <5'  sin  a/  +  (7  [ 

/sin  S  =  sin  S'  +  C  tan  e  J 

These  equations  may  be  solved  like  those  for  parallax.     By 
cross  -multiplication  of  the  first  two  by  sin  oc'  and  cos  oc',  we  find 
/cos  S  sin  Aaa  =  (7  cos  a'  +  D  sin  a'  =  &  sin  (#+  a')  j 
/  cos  $  cos  Aaa  =  cos^/+(7sinc<./—  I)  cos  a/  l>    ...(34) 


where  we  put,  for  the  aberration  in  R.A., 

Aaoc  =  CL  —  CL'. 
Forming  the  quotient  of  these  equations  : 

.  h  sin  (//  +  a")  sec  cT 

tan  Aaoc  =  -  —  l  —  ^—TTf  —  ^  -  v 
1  —  7t  cos  (#  +  a  )  sec  S 


302  EEDUCTION  TO  APPARENT  PLACE  [§  158. 

For  the  declinations  we  add  the  products  of  the  equations  (34) 
by  sin  J  Aaa  and  cos  J  Aaoc  respectively,  thus  obtaining 

/cos  S  cos  £  Aaoc  =  cos  S'  cos  \  A0a-  h  cos  (H+a.'+  \  A0an     ,og\ 
/cos  5  =  cos  (5'  -  /t  cos  (# + a'  +  J  A0oc)  sec  \  Aaoc  J 

Then,  by  cross-multiplication  of  this  equation  and  (33)3  by 
sin  S  and  cos  S,  and  putting 

we  have 

/sinAa(5=Otan€Cos^+Asin^cos(JEr+a/  +  JAaa)secJAaa, 

/cos  Aa(S  =1  +  0  tan  e  sin  S'  —  h  cos  tf  cos  (//"  +  OL  +  J  Aaoc)  sec  £  A0oc. 

If  we  compute  ^  and  «/  from 

,;'  sin  J=  C  tan  e  t  C37V 

j  cosJ=h  cos  (F+oc'4-  J  Aaa)  sec  \  AaaJ ' 

the  quotient  of  these  equations  will  give 


.(38) 


The  equations  (31),  (35),  and  (38)  give  the  reduction  for 
nutation  and  aberration  respectively.  It  is  to  be  noted  that  in 
(31)  the  oc  and  S  with  which  we  start  are  the  mean  coordinates, 
while,  in  (35)  and  (38),  they  are  the  coordinates  affected  by 
nutation.  We  may,  without  any  drawback,  reverse  the  order 
of  the  two  corrections,  computing  the  aberration  with  the  mean 
place  of  the  star,  and  then  the  nutation  with  the  place  as 
affected  by  aberration.  As  a  check  upon  the  accuracy  of  the 
work  it  may  be  well  to  make  the  computation  in  both  these 
orders. 

Section  III.    Practical  Methods  of  Reduction. 

159.  Although  the  preceding  exposition  of  the  methods  of 
reduction  is  complete  so  far  as  the  theory  of  the  work  is 
concerned,  it  it  necessary  to  minimize  the  labour  of  applying 
the  theory  by  making  the  best  use  of  the  data  in  the  ephemeris, 
and  omitting  all  processes  which  are  not  necessary  to  the  special 
problem  in  hand.  The  astronomical  ephemerides  give  not  only 
the  day  numbers  for  each  day  in  the  year,  but  ephemerides  of 


§  160.]  PRACTICAL  METHODS  OF  REDUCTION  303 

the  apparent  places  of  several  hundred  fundamental  stars,  which 
will,  in  all  ordinary  cases,  relieve  the  astronomer  from  the 
necessity  of  making  any  computations  relating  to  the  apparent 
places  of  these  particular  stars.  But  when  an  unusual  degree 
of  theoretical  precision  is  required  in  the  results,  there  are 
certain  points  which  require  attention  even  in  using  the 
ephemeris  for  this  purpose.  There  is,  in  fact,  when  labour- 
saving  devices  are  applied,  a  practical  difficulty  arising  from 
the  periods  and  values  of  the  terms  of  nutation.  These  terms 
are,  in  §  134,  divided  into  three  classes  according  to  the  length 
of  their  period.  In  the  case  of  the  larger  terms,  the  period  is 
that  of  the  moon's  node,  18'6  y.,  or  its  half.  Next  in  the  order, 
both  of  length  of  period  and  of  magnitude,  are  the  annual  or 
semi-annual  terms. 

Neither  of  these  classes  of  terms  offers  any  difficulty  growing 
out  of  the  length  of  period.  The  difficulty  arises  in  dealing 
with  the  small  terms  of  the  third  class,  the  length  of  whose 
periods  is  about  a  month  or  some  fraction  of  a  month.  The 
largest  of  these  is  within  the  limit  of  error  of  all  but  the  most 
refined  observations,  but  not  far  enough  within  to  be  always 
neglected  as  unimportant.  The  method  of  dealing  with  them 
will  be  seen  by  a  survey  of  the  practical  conditions  and  data 
of  the  problem. 

160.  Treatment  of  the  small  terms  of  nutation. 

The  astronomical  ephemeris  gives  the  apparent  positions  of 
the  principal  fixed  stars  to  O'Ol  s.  in  R.A.  and  0"'l  in  Dec.  for 
every  tenth  day  of  the  year.  In  the  case  of  the  close  polar 
stars  the  positions  are  given  for  every  day. 

In  the  ten-day  ephemeris  it  would  be  useless  to  include  the 
terms  of  short  period,  because  an  interpolation  of  such  terms  to 
intermediate  dates  could  not  be  made  with  accuracy.  We 
readily  see  that,  where  the  period  of  the  term  is  14  days,  the 
term  might  be  negative  at  two  consecutive  ten-day  epochs,  and 
pass  through  its  maximum  positive  value  during  the  interval. 
It  follows  that  when  the  astronomer  makes  use  of  the  ten-day 
ephemeris  he  must  ignore  these  short-period  terms  altogether, 


304  REDUCTION  TO  APPARENT  PLACE  [§  160. 

or  spend  much  labour  in  applying  them.  Moreover,  when,  as  is 
the  custom,  they  are  included  in  the  positions  of  the  polar  stars, 
but  omitted  from  those  of  other  stars,  there  is  a  non-homogeneity 
in  the  results  which  may  be  productive  of  confusion. 

We  begin  a  more  special  study  of  the  conditions  by  noting 
that  the  terms  of  nutation  in  KJL,  which  are  larger  than  those 
in  Dec.,  may  be  divided  into  two  classes :  those  which  vary  with 
the  declination,  having  tan<5  as  a  factor,  and  those  which,  at 
any  moment,  are  independent  of  the  declination,  and  therefore 
the  same  for  all  declinations. 

If  no  coordinates  but  equatorial  ones  were  ever  used  in 
astronomy,  the  latter  terms,  whatever  their  magnitude,  could  be 
dropped  out  as  unnecessary.  We  should  then  be  referring  all 
RA/s,  not  to  the  apparent  equinox  of  the  date,  but  to  a  quasi 
mean  equinox  affected  by  all  the  other  inequalities,  as  an  origin. 
The  reason  why  this  equinox  is  not  adopted  as  the  origin  of 
RA.  is  that  the  motions  of  the  planets  are  in  the  first  place 
necessarily  referred  to  the  ecliptic  as  the  fundamental  plane; 
and,  in  order  to  obtain  a  correct  reduction  to  the  equator,  the 
actual  equinox  at  each  day,  with  all  its  inequalities,  must  be 
made  use  o£  It  is  quite  possible  that  if,  following  this  practice 
so  far  as  the  original  computations  were  concerned,  the  system 
were  universally  adopted  of  dropping  constant  terms  of  nutation 
from  the  KJL  of  all  heavenly  bodies,  using  them  only  in  the 
original  computations  where  longitudes  entered,  it  would  be  a 
simplification  of  our  present  system,  which  would  carry  with  it 
no  serious  drawbacks. 

No  such  scheme  is,  at  present,  practicable  in  its  entirety.  But 
at  a  conference  held  in  Paris  in  1896,  at  which  the  Directors 
of  the  principal  astronomical  ephemerides  devised  a  uniform 
system  of  dealing  with  star-reductions,  it  was  agreed  to  drop 
from  the  RA.  of  all  stars  those  minute  constant  terms  of  short 
period  which  are  common  to  all  the  stars.  A  step  is  thus  taken 
toward  the  simplification  which  has  been  suggested  in  the 
origin  of  Right  Ascension. 

Although  we  thus  get  rid  of  those  parts  of  those  nutation  terms 
of  short  period  which  are  common  to  all  the  stars,  we  do  not 


5  16L]  TREATMENT  OF  THE  SMALL  TERMS  OF  NUTATION    305 

thereby  avoid  the  terms  which  vary  with  the  declination.  The 
celestial  pole  does  actually  go  through  two  revolutions  per 
month  in  a  very  small  curve  0"'17  in  diameter,  approximating 
to  a  circle ;  and  our  instruments,  being  carried  upon  the  moving 
earth,  are  affected  by  this  motion,  which  must  therefore  be 
taken  account  of  in  the  most  refined  reductions.  A  small 
correction  depending  on  the  tangent  of  the  declination  is  there- 
fore included  in  the  ephemerides  of  the  polar  stars.  This  gives 
rise  to  a  non-homogeneity  between  the  star  positions  given  for 
every  ten  days  and  those  given  for  every  day. 

The  terms  in  question  are  so  minute  that  the  practical 
astronomer  has,  in  all  ordinary  cases,  no  occasion  to  trouble 
himself  with  them.  He  can  use  the  numbers  of  the  ephemeris 
with  entire  confidence  that  he  will  be  led  into  no  appreciable 
error  by  the  lack  of  homogeneity.  If  engaged  in  any  special 
research  in  which  so  small  a  correction  is  important,  the 
ephemeris  supplies  all  data  necessary  for  his  purpose. 

161.  Development  of  the  reduction  to  terms  of  the  second  order. 

Although  the  computation  of  the  reduction  by  the  preceding 
rigorous  formulae  will  probably  be  found  simpler  than  the  use 
of  a  development  in  series,  when  only  a  single  reduction  is 
wanted,  there  are  some  purposes  in  which  a  development  of  the 
reduction  is  required.  Unless  the  star  to  be  reduced  is  within 
b'  of  either  pole,  a  development  to  terms  of  the  second  order  will 
be  sufficient.  When  we  drop  terms  of  the  third  and  higher 
orders  in  the  development,  a  number  of  simplifications  may  be 
made  in  the  process  by  dropping  out  all  terms  which,  in  the 
final  result,  will  rise  only  to  the  third  order.  The  following  are 
some  of  the  cases  in  which  this  or  other  simplifications  may  be 
made : 

1.  Since  the  tangent  of  a  small  arc  differs  from  the  arc  itself 
only  by  a  quantity   of  the   third   order,  it   follows  that,  in 
developing  to  terms  of  the  second  order,  we  may  substitute  the 
reduction  itself  for  its  tangent. 

2.  For  the  same  reason  the  cosine  and  secant  of  a  quantity  of 

the  order  of  magnitude  of  the  reduction  may  be  supposed  equal 
X.S.A.  u 


306  REDUCTION  TO  APPARENT  PLACE  [§  161. 

to  unity,  and  therefore  dropped  as  a  factor  when  multiplied  by 
the  reduction  itself. 

3.  In  forming  the  several  increments  of  the  reductions  of  the 
first  order,  in  order  to  obtain  the  terms  of  the  second  order,  it 
will  be  sufficient  to  carry  these  terms  to  the  first  order  only. 

4.  So  far  as  the  terms  of  the  second  order  are  a  function  of 
the  coordinates  of  the  star,  it  is  indifferent  whether  we  use  the 
mean  or  apparent  values  of  these  coordinates  in  the  expressions 
for  such  terms. 

5.  For  the  reason  already  mentioned,  the  only  terms  of  the 
second  order  which  need  to  be  included  in  the  R.A.  are  those 
which   contain   terms   of  two  dimensions  in  secS  or  tan 3.     In 
the  case  of  the  declination  all  terms  may  be  dropped  which  do 
not  contain  either  tan  S  or  sec  S  as  a  factor. 

In  forming  the  required  increments  of  the  second  order  it  will 
be  our  object  to  first  express  them  in  terms  of  g,  G,  h,  etc.,  and 
then  replace  these  quantities  by  their  expressions  in  terms  of  the 
Besselian  day  numbers  A,  B,  C,  D,  by  means  of  the  equations 
(26),  (27),  and  (28). 

Following  the  same  order  as  in  the  preceding  rigorous  reduction, 
we  shall  begin  by  forming  the  terms  of  the  second  order  due  to 
precession  to  date  and  nutation  alone,  which  terms  we  shall 
designate  by  the  symbols 

AX  A^. 

The  terms  of  the  second  order  due  to  aberration  will  then  be 
found  by  assigning  the  increments  A7loc,  AW<S  of  the  first  order  to 
the  expressions  for  the  reduction  for  aberration,  and  also  the 
increments  consisting-  of  the  terms  of  the  first  order  in  the 

O 

aberration  itself.  The  aberration-terms  of  the  second  order 
will  then  be  the  changes  in  the  aberration  due  to  these  increments 
of  the  first  order.  The  combined  increments  of  the  second  order 
thus  arising  will  be  designated  as 

A^a,  A|,a<5,  AX  A» 

162.  Precession  and  nutation. 

Beginning  with  the  terms  arising  from  precession  to  date  and 
nutation  combined,  we  write  the  necessary  portions  of  the 


§  163.]  TERMS  OF  THE   SECOND  ORDER  307 

rigorous  reduction  as  given  in  (31)  in  the  following  form,  where 
we  have  substituted  for  a  its  value  a0+  G  : 


l-2/cos(G+a0)         ................  (40) 


Neglecting  p'cos(6r-f  oc0)  in  the  denominator  of  the  fraction, 
the  expression  for  Anoc  will  reduce  to  the  reduction  already  found 
for  terms  of  the  first  order.  When  terms  of  the  second  order 
only  in  pf  are  included,  we  may  write 


Thus  the  terms  of  the  second  order  in  the  reduction  of  the 
right  ascension  become 


=  (B  cos  a  +  An  sin  oi)(An  cos  a  —  B  sin  oc)tan2(5  [~   •  •  -(41) 
=  J  {ABn  cos  2a.+(AW  -  £2)siii  2a}  tan2<? 

For  the  corresponding  terms  in  the  declination  we  have 


By  easy  reductions  this  becomes 

A^  =  {  -i#2  +  K,4V2-£2)cos  2aL-±ABnsm  2oc}  tan  8.   (42) 

163.  Aberration. 

Passing  now  to  the  aberration  :  in  order  to  obtain  its  complete 
effect  we  have  to  substitute  for  a.'  and  8'  in  (35)  and  (38)  the 
values  oc0  +  Anoc  and  $0+  An&  We  also  have  to  include  the  terms 
of  the  second  order  resulting  immediately  from  the  development 
of  the  denominator.  The  latter  are,  for  the  R.A.  : 

A2aoc  =  k2  sin  (  H  +  a)  cos  (H  +  a)  sec2£ 

=  [CD  cos  2oc+  J(^2-  G2)  sin  2a}  sec2& 

Here,  as  before,  we  use  the  symbols  oc  and  S  without  farther 
specification,  because  the  terms  are  of  the  second  order. 

For  the  substitution  of  Artoc  and  AnS  we  require  the  ex- 
pressions (35)  and  (38)  to  the  first  order  only,  using 

Aaoc  =  h  sin  (H+  GCO  +  Anoc)  sec  (<S0  +  An<5). 


308  EEDUCTION  TO   APPARENT  PLACE  [§  163. 

Then 
A*,aa.=  h  cos  (F+a)  Anoc  sec  S  +  h  sin  (H+a.)  An<$  sec  8  tan  6.  (43) 

We  assign  to  Anot  and  A;lcJ  their  values  (40),  taken  only  to 
terms  of  the  first  order,  namely 

Anoi  =  g  sin  (  G  +  a)  tan  8  =  (  An  sin  a.  -f  B  cos  a)  tan  8, 
An<S  =  </  cos  (G  -f  «.)  =  ^  cosoi  —  B  sin  a. 

These  increments  being  substituted  in  (43),  the  latter  reduces  to 

A|,aa=^  sin  (6r  +  H+  2a)  tan  <$  sec  & 
The  factor  of  tan  8  sec  <?  being 


we  find  that 

=  A  Cn  +  BD, 


We  now  obtain 

Ai,a«.  =  {(A  Cn  +  -BD)  cos  2a  +  (ADn  -  BC)  sin  2oc}  tan  3  sec  A  (44) 
Proceeding  in  the  same  way  with  the  declination,  we   find 
that  the  terms  of  the  second  order  in  (38)  are  found  by  writing 
the  latter  in  the  form 


=   sn 
and  are,  when  aberration  only  is  considered, 

Ay=j2sin(J+^)cos(J+(5)  ...................  (45) 

Comparing  with  (37),  we  see  that  neither  factor  of  this 
product  is  increased  in  approaching  the  pole  ;  A^  may,  therefore, 
be  dropped,  leaving  only  the  nutational  increment  of  (38)  to  be 
considered.  We  reduce  the  principal  term  of  (38)  thus  : 

Aa,5=.;sin(/+<5')  \  ,46) 

=j  sin  /  cos  #  +J  cos  J  sin  8')' 

Beginning  with  the  aberrational  increment  of  this  expression, 
we  see  from  (37)  that  sin  J  does  not  contain  a  or  8.  For  the 
increment  of  cos  J  we  have 

A00'cos/)=  —  JAsin(£r+a)  Aaoc. 
From  (35),  the  principal  value  of  Aaoc  is 

Aaa  =  h  sin  (H  +  a)  sec  8. 


§  164.]  TEEMS   OF  THE   SECOND   OEDER  309 

We  have  from  (27), 

h  sin  (H-{-  oi)  =  C  cos  a.  +  D  sin  a.. 

By  these  successive  substitutions  the  increment  of  (46)  becomes 
A2a<$  =  -{i(C"2  +  -D2)  +  i(C'2-D2)cos2a  +  JC'Dsm2(x}  tan  5.  (47) 
We   next   assign  to  cos  J  its  nutational   increment  through 
putting  in  (37) 


We  thus  have 

AnO'  cos  J)=  —h  sin  (H+  OL)  A^oc 

=  —  (G  cos  a  +  D  sin  a)  Ana. 

By  substitution  of  Ana,  the  increment  of  (46)  reduces  to 


(48) 

To  complete  the  expression  for  Aa,»<$,  the  nutation  increment 
has  to  be  applied  to  $'  in  (46).  But  this  increment  does  not 
increase  toward  the  pole,  and  may  therefore  be  dropped. 

164.  It  is  important  to  know  how  near  the  pole  these  terms 
become  important.  For  this  purpose  we  remark  that  the  values 
of  An,  B,  C,  and  D  are  commonly  less  than  20",  or  *0001.  The 
coefficients  of  the  terms  of  the  second  order  are  therefore  nearly 
always  less  than  0"'002. 

Near  the  pole,  and  at  a  distance  of  p°  from  it,  the  values  of 

57° 
tan  S  and  sec  S  do  not  differ  greatly  from  —5-.      We   therefore 

have  the  following  approximations  to  values  which  the  terms  of 
the  second  order  will  never  greatly  exceed : 

p°  =  12°;          A2  =  0"-0r 
6  0-02 


.(49) 


„        4  „      0-03 

» ""•   3  „     0  -04 

„        2  „      0-06 

»        1  ,,      0  -12, 

We  conclude  that  when  only  the  ordinary  limit  of  precision, 

±0"'l,  is  aimed  at,  the  trigonometric  reduction  need  be  used  only 


310  KEDUCTION  TO  APPAKENT  PLACE  [§  164. 

within  2°  of  the  pole,  but  that,  if  the  limit  is  to  be  +  0"'01, 
either  it  or  the  development  to  terms  of  the  second  order 
should  be  used  within  12°  of  the  pole. 


Section.  IV.    Construction  of  Tables  of  the  Apparent 
Places  of  Stars. 

165.  The  term  fundamental  is  applied  to  a  limited  number  of 
the  best  determined  stars,  the  known  positions  of  which  are  used 
as  auxiliaries  to  determine  the  positions  of  all  other  heavenly 
bodies.  There  is  no  definable  limit  to  the  number  of  stars  that 
may  be  used  for  this  purpose.  About  the  beginning  of  the 
19th  century  Maskeleyne  chose  thirty-six  of  the  brightest  stars, 
nearly  all  of  the  first  or  second  magnitude,  scattered  over  that 
portion  of  the  sky  which  could  be  seen  at  Greenwich,  made 
frequent  observations  upon  them,  and  thus  determined  their 
positions  with  all  the  accuracy  of  which  his  instruments  per- 
mitted. These  were  used  to  determine  the  positions  of  other 
stars  by  methods  the  principles  of  which  will  be  shown  in  the 
next  chapter.  Since  his  time  the  increasing  requirements  of 
astronomy  have  led  to  a  continual  increase  in  the  number  of 
stars  regarded  as  fundamental. 

In  the  preparation  of  each  of  the  national  astronomical 
ephemerides,  a  selection  of  stars  to  be  regarded  as  fundamental 
has  been  made,  and  their  apparent  places  given  on  the  plan  set 
forth  in  the  preceding  section.  Quite  independently  of  these 
lists,  other  lists,  generally  more  numerous,  have  been  prepared 
from  time  to  time  by  astronomers  for  special  purposes.  The 
most  complete  list  of  this  kind  is  found  in  the  Astronomical 
Papers  of  the  American  Ephemeris,  vol.  viii.,  pp.  91-122.  It 
comprises  all  the  stars  whose  places  are  given  in  any  of  the 
astronomical  ephemerides,  with  the  addition  of  such  other  stars 
that  the  whole  shall  form  a  list  scattered  over  the  entire  sky 
with  as  near  an  approach  to  uniformity  as  possible.  The  number 
of  stars  in  this  list  is  1597.  This  number  is  more  than  double 
that  thus  far  given  in  any  one  ephemeris.  But  it  is  not  unlikely 
that  at  no  distant  date  arrangements  may  be  made  between 


S  166.]   TABLES   OF  THE  APPARENT  PLACES   OF   STAES         311 

these  publications  which  shall  admit  of  the  apparent  places  of 
the  entire  list  of  selected  stars  being  regularly  given. 

When  apparent  places  of  such  stars  as  these  are  required,  not 
merely  for  a  single  year,  but  for  a  number  of  consecutive  years, 
their  computation  can  be  facilitated  by  the  use  of  suitable  tables, 
by  a  method  devised  by  Bessel  and  found  in  his  Tabulae  Regio- 
montanae.  The  method  of  constructing  such  tables  will  here  be 
set  forth  in  such  a  way  that  its  application  need  not  involve  any 
difficulty  to  one  acquainted  with  the  subject. 

166.  The  fundamental  idea  of  the  method  is  that  the  day 
numbers,  omitting  the  minute  terms  of  short  period,  are  functions 
of  the  longitude  of  the  moon's  node  and  of  the  sun's  longitude. 
They  may,  therefore,  be  divided  into  two  parts  depending  on 
these  two  arguments. 

The  star  numbers  varying  slowly  from  year  to  year,  it  follows 
that  their  products  into  the  day  numbers,  the  sum  of  which 
products  is  the  reduction  to  apparent  place,  may  be  arranged,  in 
the  case  of  each  star,  into  two  tables,  one  depending  on  the  node 
and  the  other  on  the  sun's  longitude,  or  the  time  of  year.  To 
show  the  process  more  in  detail,  we  put 

Tlt  the  time  corresponding  to  the  beginning  of  any  solar  year  ; 

«.1;  Sv  the  mean  coordinates  of  the  star  for  the  beginning  of 
that  year  ; 

T,  the  fraction  of  the  year  after  the  beginning  at  which  the 
apparent  position  is  required. 

Let  us  now  see  how  the  reduction  from  the  mean  place  for  Tl 
to  the  apparent  place  for  T^  +  T  may  be  tabulated.  By  the 
developments  in  Chapter  X.,  §§  145,  146,  the  change  due  to  pre- 
cession and  proper  motion  from  the  beginning  of  any  solar  year 
until  the  date  r  may  be  expressed  in  the  form 


The  speed  of  variation  Dtoi  is  computed  numerically  by  the 
formulae  of  §  145.  The  term  D\GL  will  never  be  required  except 
when  the  star  is  near  the  pole.  Whether  it  is  used  or  not,  the 
values  of  the  two  factors  change  from  year  to  year  only  with 


312  REDUCTION   TO  APPARENT  PLACE  [$  166. 

great  slowness.  We  may,  therefore,  compute  their  values 
numerically  for  some  fundamental  epoch,  say  1900,  and  also 
their  secular  variations.  The  latter,  at  least  in  the  case  of  D-(OL 
and  D2tS,  may  most  easily  be  found  by  repeating  the  computation 
for  a  second  epoch,  say  1925  or  1950.  In  this  way  we  shalf 
derive,  for  each  star,  a  general  expression  of  the  form 


with  similar  expressions  for  D]a.,  DtS,  and  D2tS.  We  then  make 
a  tabJe  of  Axa  and  A^  from  (50)  for  intervals  of  ten  sidereal 
days,  or,  if  the  star  is  near  the  pole,  for  every  such  day.  If  a 
ten-day  interval  is  used,  the  increment  of  T  from  one  date  to  the 
next  will  be  -i  rv 


Starting  the  table  from  the  beginning  of  the  year,  the  suc- 
cessive values  of  T  and  AjO.  will  be 

r  =  0;         AT;          2Ar....] 

Al0c  =  0;      oc/Ar;      2a'Ar....    .............  (51) 

sec.  var.  =  0  ;     a"Ar  ;     2oc//Ar  .  .  .  .  J 

The  products  Jr2-D?oc  may  be  included  in  AjO.  in  the  exceptional 
case  when  they  are  sensible.  Thus,  from  a  single  table  com- 
prising an  argument  and  two  columns  may  be  interpolated  the 
value  of  AjO.  for  1900  and  its  secular  variation  for  any  date  r. 
By  multiplying  the  secular  variation  by  the  fraction  of  a 
century  elapsed  since  the  fundamental  epoch,  and  adding  the 
product  to  \OL  for  1900,  the  complete  value  of  AjOC  for  the  year 
required  will  be  obtained. 

Of  course  the  tabular  value  of  Ax<5  may  be  formed  in  the  same 
way. 

Nutation.  Passing  to  the  nutation,  we  use  the  equations  (3) 
of  §  151.  The  four  coefficients 

cos  e  +  sin  e  sin  a  tan  8\ 

cos  octant,  /52) 

cos  oc  sin  e, 

since, 


§  166.]  TABLES  OF  THE  APPARENT  PLACES  OF  STAES    313 

are  nearly  constant  for  any  one  star,  being,  in  fact,  in  the  case 
of  the  second  and  fourth  of  these  coefficients,  identical  with 
b  and  —  ?>',  while  the  first  and  third  are  nearly  equal  to  a  and  a'y 
multiplied  by  constant  factors. 

The  values  of  A^  and  Ae  by  which  these  four  coefficients  are 
to  be  multiplied  are  found  in  §  134,  where  their  terms  are  divided 
into  three  classes,  which  are  to  be  tabulated  separately  as  follows : 

Terms  depending  on  the  node.  Calling  T^  the  time  of  be- 
ginning of  the  year,  we  compute  the  value  of  0,  the  longitude 
of  the  moon's  node,  for  the  five  dates  : 

7\ ;     2\  +  100  sid.  days ;     2^  +  200  sid.  days,  etc., 
or      2\;     ^  +  0-273;     Z\  + 0-546;     2^  + 0-819;     Z\  + 1-092, 
which  carries  the  computation  past  the  end  of  the  year.     With 
these  values  of  Q  the  terms  of  Ai/r  and  Ae  depending  on  Q  are 
computed  and  multiplied  by  the  four  corresponding  numbers  (52). 
This   computation   may   readily   be   made    for   every   year   for 
which  tables  are  to  be  used.     We  thus  obtain  so  much  of  the 
reduction  to  apparent  place  as  depends  on  the  longitude  of  the 
node,  which  we  may  designate  by  the  symbols  Af2a  and  AQ*. 

On  Bessel's  plan,  the  values  of  a0  and  S0  are  added  to  these 
terms  in  the  printed  table,  for  which  we  then  have  five  values 
for  each  year.  From  these  the  values  for  any  intermediate  date 
may  be  interpolated. 

Annual  terms.  In  these  the  nutation  terms,  which  belong  to 
the  second  class,  are  combined  with  the  aberration,  both  being 
functions  of  the  sun's  true  longitude.  Since  r  =  0  when  the 
sun's  mean  longitude  is  280°,  it  is  easy  to  tabulate  the  values 
of  0  for  T  =  ()5  T==AT,  r=2Ar,  etc., 

through  the  year.  These  values  of  0  are  then  used  in  com- 
puting the  corresponding  values  of  Ai/>-  and  Ae,  and  also  of  C 
and  D  from  (15).  By  multiplying  each  by  the  proper  coefficients, 
which  are  functions  of  the  position  of  the  star,  so  much  of  the 
reduction  to  apparent  place  as  depends  on  the  sun's  longitude 
is  thus  tabulated  as  a  function  of  r.  To  the  quantities  thus 
tabulated  are  then  added  the  values  of  A^  and  A^  for  the 
corresponding  values  of  T  from  (51). 


314  KEDUCTION   TO  APPARENT  PLACE  [§  166. 

The  secular  variation  of  this  part  of  the  reduction  may  be 
computed  and  applied  on  the  same  principle  with  that  of  the 
precession. 

From  the  four  tables  thus  formed,  two  for  each  coordinate, 
we  may  form  the  apparent  R.A.  and  Dec.  of  the  star  for  any 
time  T  in  any  year.  We  now  have  to  show  how  the  dates  of 
the  year  are  related  to  the  meridian  of  the  place  for  which  an 
ephemeris  of  the  star  may  be  required. 

167.  Since  the  apparent  places  of  fundamental  stars  are 
required  almost  entirely  in  connection  with  observations  across 
the  meridian  of  some  observatory,  the  ephemerides  give  these 
places,  not  for  mean  noon,  but  for  the  moment  of  passage  over 
the  meridian  of  the  principal  observatory  for  which  the  ephe- 
merides are  constructed.  Since  the  interval  between  two 
transits  is  a  sidereal  day,  the  tables  are  constructed  for  units  of 
an  integral  number  of  sidereal  days,  and  not  for  mean  time.  It 
follows  that,  for  any  one  observatory  and  any  one  year,  the 
factor  of  interpolation,  omitting  the  entire  days,  will  be  the 
same  for  any  star  through  the  whole  course  of  any  one  year. 
This  factor  depends  on  the  relation  of  the  beginning  of  the  year 
to  the  meridian  of  the  observatory  in  question. 

The  moment  1\  at  which  the  solar  year  begins,  being  that  at 
which  the  sun's  mean  KA.  is  18  h.  40  m.,  is  a  certain  moment 
of  absolute  time,  of  which  the  expression  in  the  local  time  of 
.any  place  will  depend  on  the  longitude  of  that  place.  There 
will  be  one  meridian,  and  no  more,  at  which  the  sidereal  time 
of  Tj  will  be  18  h.  40  m.  This  meridian  is  that  which  the  mean 
sun  crosses  at  the  moment.  It  follows  that  this  moment  is 
mean  noon  on  this  meridian.  Let  us  call  the  latter  the  standard 
meridian  for  the  year.  Let  us  put 

k,  the  east  longitude  of  the  standard  meridian  from  Greenwich. 

It  follows  that  the  Greenwich  mean  time  of  the  beginning  of 
the  solar  year  is  —k  or  24  h.  —  k.  This  time  may  be  computed 
from  year  to  year  by  tables  of  the  sun's  mean  longitude.  In 
Appendix  III.  of  the  present  work  will  be  found  a  table  of  the 
times  for  the  twentieth  century.  Changing  the  signs  of  these 


§  167.]    TABLES  OF  THE  APPARENT  PLACES  OF  STARS          315 

times,  or  subtracting  them  from  one  day,  and  converting  the 
result  into  hours  and  minutes  if  required,  we  shall  have  the 
values  of  k.  Next,  let  us  put 

X,  the  west  longitude  from  Greenwich  of  the  meridian  =M 
for  which  the  ephemeris  is  required. 

\-\-k  will  then  be  the  distance  west  from  the  standard  meri- 
dian to  the  meridian  M  in  question.  It  follows  that  on  this 
meridian  we  shall  have  at  the  moment  of  beginning  of  the 
solar  year 

Local  mean  time  =  24  h.  —  (X  +  k). 
Sidereal  time       =  18  h.  40  m.  -  (X  +  k). 

If -we  express  X  and  k  in  fractions  of  a  day,  the  first  transit 
of  the  vernal  equinox  over  the  meridian  M  in  the  course  of  any 
one  year  will  occur  at  a  sidereal  interval  5  h.  20  m.  +  X  +  &  after 
the  beginning  of  the  year. 

Also,  the  first  transit  of  a  star  of  R.A.  =  a  over  this  meridian 
after  the  beginning  of  the  year  will  follow  the  beginning  by 
the  sidereal  interval 

5  h.  20  m.  +  oc+X+/c  —  (24  h.  when  necessary). 

This  then  will  be  the  factor  of  interpolation  for  the  first 
transit.  The  factor  for  any  subsequent  transit  will  be  that 
corresponding  to  an  integral  number  of  days  after  this  moment. 

NOTES  AND  REFERENCES. 

The  large  scale  on  which  reductions  from  mean  to  apparent  positions  of 
stars,  or  the  reverse,  have  to  be  made,  has  led  to  a  number  of  methods  and 
tables  auxiliary  to  the  regular  ones  of  the  ephemeris  which  alone  have  been 
treated  in  the  present  chapter. 

STONE,  E.  J.,  Tables  for  facilitating  the  computations  of  star  constants 
v( Appendix  to  Cape  Observations  1874),  gives  extended  tables  for  the  easy 
and  rapid  computation  of  the  star  constants,  a,  b,  c,  and  d. 

The  Struve-Peters  values  of  the  astronomical  constants  were  not  intro- 
duced into  the  ephemerides  until  about  1850.  In  order  to  facilitate  their 
use  before  that  time  the  Poulkova  Observatory  computed  and  published 
tables  of  the  day-numbers  under  the  title  Tabulae  Quantitatum  Besselianarum 
for  the  period  1750-1894.  The  later  publications  of  the  series  differ  from 
those  given  in  the  ephemerides  by  including  the  smaller  terms  of  the 


316  REDUCTION  TO  APPARENT  PLACE 

nutation.  The  numbers  are  therefore  given  for  every  day.  Before  1840 
they  are  given  only  for  every  tenth  day. 

AUWERS,  Tafeln  zur  Reduction  von  Fixstern-Beobachtungen  fur  1726-1750 
(Zweites  Supplementheft  zur  Vierteljahrsschrift  der  astronomischen  Gesell- 
schaft,  Jahrgang  IV.),  Engelmann,  Leipzig,  1869,  gives  the  day  numbers 
for  ten-day  intervals  with  modern  values  of  the  constants,  thus  forming, 
with  the  Poulkova  series,  a  complete  series  from  1726. 

A  modification  of  the  independent  day  numbers,  and  of  the  method  of 
using  them,  has  been  devised  by  Mr.  W.  H.  Fin  lay,  assistant  at  the  Cape 
Observatory,  which  is  believed  to  offer  marked  advantages  over  the  usual 
ones,  still  tabulated  in  the  ephemeris.  The  system  is  explained  in  Monthly 
Notices  of  the  Royal  Astronomical  Society,  vol.  1.,  p.  497,  (June  1890).  Tables 
of  the  star  constants  used  in  this  method  have  been  published  by  the  Cape 
Observatory,  and  the  modified  day  numbers  for  use  in  connection  with  them 
have  been  published  by  the  same  institution  since  1897. 

Although  the  logarithms  of  the  day  numbers  are  given  to  four  decimals  in 
the  ephemeris,  three  are  practically  sufficient  in  the  reduction  of  meridian 
observed  positions  to  mean  place,  unless  near  the  pole.  Tables  of  3-place 
logarithms  are  found  at  the  end  of  the  present  volume. 

Several  graphical  systems  for  the  reduction  have  been  devised.  One  of 
these  is  by  Mr.  Finlay  and  is  found,  with  the  chart  for  its  application,  in 
Monthly  Notices,  R.A.S.,  vol.  lv.,  p.  15,  (November  1894).  Another  by 
Erasmus  D.  Preston  is  found  in  Bulletin  of  the  Philosophical  Society  of 
Washington,  vol.  iii.,  p.  182.  This  was  independently  distributed  by  the 
Society.  It  contains  a  diagram  for  finding  the  reduction  graphically 
without  the  labour  of  computation. 

The  fact  that  a  ten  day  ephemeris  of  a  star  for  an  entire  year  can  be 
computed  from  tables  of  the  form  described  in  the  present  chapter  in  about 
an  hour  renders  the  use  of  such  tables  desirable  in  the  computations  of 
annual  ephemerides  extending  through  a  number  of  years.  Besides  those  of 
the  Tabulae  Regiomontanae  which  are  based  on  the  older  values  of  the 
constants,  tables  for  the  fundamental  stars  by  Leverrier  are  found  in  Annals 
de  I  Observatoire  de  Paris  ;  Memoires,  vol.  ii.  Tables  for  a  larger  number  of 
stars,  slightly  different  in  some  details,  are  found  in  Star  Tables  of  the 
American  Ephemeris,  Nautical  Almanac  Office,  Washington. 

In  the  office  of  the  British  Nautical  Almanac  the  reductions  are  under- 
stood to  be  computed  for  each  separate  day  by  the  use  of  a  "  Star  correction 
Facilitator,"  an  ingenious  instrument  devised  by  Mr.  T.  C.  Hudson,  and 
described  in  the  Monthly  Notices,  R.A.S.,  liv.,  p.  90,  (December,  1893). 


CHAPTER  XII. 

METHOD  OF  DETERMINING  THE  POSITIONS  OF   STARS 
BY  MERIDIAN  OBSERVATIONS. 

168.  In  this  branch  of  practical  astronomy  everything  relating 
to  the  management  of  the  instrument,  and  the  investigation  of 
its  performance,  belongs  to  the  subject  of  practical  and  instru- 
mental astronomy,  to  be  treated  in  another  work.  In  the 
present  work  we  shall  develop  the  general  principles  which  enter 
into  the  determination  of  positions  of  the  fixed  stars  from 
observations. 

Such  determinations  are  divided  into  two  classes,  fundamental 
and  differential. 

Fundamental  work  consists  in  the  determination  of  positions 
of  fixed  stars,  the  results  of  which  are  independent  of  any 
previous  determinations. 

Differential  work  is  that  in  which  positions  previously  deter- 
mined are  assumed  as  known,  and  new  positions  are  fixed 
relatively  to  these  assumed  ones. 

Even  in  fundamental  work  it  is  nearly  always  advisable,  at 
least  in  the  case  of  right  ascensions,  to  assume  certain  positions 
of  the  stars  in  advance,  with  the  view  of  subsequently  correcting 
them  from  observations  in  such  a  way  as  to  obtain  results 
independent  of  any  preceding  work.  Results  of  this  kind  are  to 
be  regarded  as  fundamental,  leaving  as  differential  only  results 
in  which  preceding  determinations  are  regarded  as  not  subject  to 
correction. 


318  OBSERVED  POSITIONS   OF   STARS  [§  169. 

169.  The  ideal  transit  instrument. 

The  development  of  the  subject  requires  a  conception  of  the 
instruments  used  in  determinations  of  position  in  the  heavens,  in 
a  state  of  ideal  perfection.  The  actual  instruments  used  by 
the  astronomer  require  a  great  number  of  small  corrections  for 
their  errors  and  deviations.  The  object  and  result  of  these 
corrections  is  to  reduce  the  results  derived  by  the  use  of  the 
instrument  to  what  they  would  have  been  were  the  latter 
ideally  perfect. 

The  instruments  necessary  for  the  determination  of  R.A.'s 
are  the  transit  instrument  and  the  sidereal  clock,  with  their 
subsidiary  appliances.  That  required  for  the  declinations  is 
a  vertical  circle.  The  transit  instrument  and  the  circle  are 
commonly  combined  into  a  single  instrument  known  as  the 
transit  circle  or  meridian  circle;  but,  owing  to  their  separate 
functions,  they  can  be  considered  separately. 

The  ideal  transit  instrument  is  a  telescope  moving  only  on  a 
fixed  horizontal  east  and  west  axis  at  right  angles  to  its  line  of 
sight,  so  that  the  latter  describes  the  plane  of  the  meridian.  In 
the  focus  of  its  object  glass  a  spider  line  is  set  at  right  angles 
both  to  the  axis  and  the  line  of  sight.  The  latter  passes  through 
the  centre  of  the  object  glass  and  of  the  spider  line.  In  the  ideal 
instrument  as  described,  the  line  of  sight  being  always  in  the 
plane  of  the  meridian,  marks  out  the  meridian  on  the  celestial 
sphere.  Observations  with  the  actual  instrument  require  a 
number  of  corrections  for  deviation  from  the  ideal  form.  Every- 
thing relating  to  these  corrections  belongs  to  the  subject  of 
practical  and  instrumental  astronomy,  which  is  not  treated  in 
the  present  volume.  What  is  essential  for  our  present  purpose 
is  only  the  conception  of  the  ideal  instrument. 

The  ideal  clock  runs  with  perfect  uniformity,  so  that  the 
correction  necessary  to  reduce  the  indication  of  its  face  to 
sidereal  time  is  a  quantity  which  varies  uniformly  with  the 
time.  This  uniform  variation,  in  the  course  of  one  day,  is  called 
the  rate  of  the  clock.  The  ideal  rate  can  be  determined  by  the 
difference  between  the  clock  times  of  transit  of  a  star  over  the 
meridian  on  two  consecutive  days. 


§  170.]    METHOD  OF  DETERMINING  RIGHT  ASCENSIONS        319 

We  have  now  to  show  how,  with  the  ideal  instrument  as 
described,  the  right  ascensions  and  declinations  of  stars  are 
ideally  determined. 


Section  I.    Method  of  Determining  Right  Ascensions. 

170.  It  will  be  most  instructive  to  begin  with  the  ideal  case  in 
which  the  right  ascensions  of  all  the  stars  are  supposed  to  be 
unknown  quantities,  whose  values  are  to  be  found  by  observa- 
tion. Since  the  right  ascensions  are  measured  from  the 
equinoxes,  and  since  the  latter  is  an  imaginary  point  defined  as 
that  at  which  the  sun  apparently  crosses  the  celestial  equator, 
right  ascensions  must  be  determined  by  comparing  the  stars 
with  the  sun.  We  therefore  observe  the  clock  times  of  transit 
of  the  sun  over  the  meridian  with  our  ideal  instrument  day  after 
day,  through  an  entire  year.  We  also  observe  the  sun's  declina- 
tion at  the  same  transits  on  a  system  which  will  be  described  in 
the  next  section. 

We  also  observe  the  clock  times  of  transit  of  a  selected  list  of 
stars,  preferably  near  the  equator,  through  the  seasons  during 
which  such  observations  can  be  made  upon  the  star. 

Assuming  the  obliquity  of  the  ecliptic  to  be  known,  the 
longitude  and  R.A.  of  the  sun  on  every  clay  of  observation 
can  be  computed  from  its  observed  declination  by  trigono- 
metric formulae  not  necessary  to  be  given  here.  It  may  be 
remarked  that  near  the  solstices,  where  the  longitude  and  K.A. 
are  near  90°  or  270°,  these  coordinates  cannot  be  determined 
accurately  from  the  declination.  But  this  is  a  practical  detail 
wrhich  need  not  interfere  with  our  ideal  proceeding.  Our  results 
will  depend  upon  observations  of  the  sun's  declination,  made 
not  too  near  the  solstices.  A  small  error  in  the  adopted  value 
of  the  obliquity  will  also  be  nearly  or  entirely  eliminated,  because 
it  will  have  opposed  effects  at  different  seasons.  Moreover,  the 
obliquity  itself  may  be  determined  from  the  observations  of 
declination. 

Let  us  put  for  the  observations  of  any  one  day: 

TG,  the  clock  time  of  transit  of  the  sun ; 


320  OBSERVED  POSITIONS   OF  STARS  [§  170. 

jPj,  T2,  TS)  ...  ,  the  clock  times  of  transits  of  any  number  of 

stars  on  the  same  day  ; 
oc0  ,  the  right  ascension  of  the  sun,  computed  from  the  observed 

declination. 

Since  the  sidereal  time  of  transit  of  the  sun  is  identical  with 
its  right  ascension  at  the  moment  of  transit,  it  follows  that,  if 
the  sidereal  clock  is  set  exactly  right,  we  should  have 


If,  as  practically  is  always  the  case,  this  equation  is  not 
exactly  satisfied,  the  difference  is  the  correction  of  the  clock, 
which  we  call  A  C: 


From  repeated  observations  of  the  sun  day  after  day,  we  have 
a  series  of  values  of  AC.  If  the  clock  were  correct  and  the 
observations  without  error,  A6r  would  vary  by  the  same  uniform 
quantity  every  day,  and  its  general  expression  would  be  of  the 

form' 


T  being  a  constant  expressing  the  daily  rate  of  the  clock. 
Practically,  we  have  to  suppose  r  a  constant  so  long  as  no 
serious  error  will  thus  arise. 

With  the  value  of  r  and  A(70  the  value  of  A(7  can  be  deter- 
mined at  the  moment  of  transit  of  every  star.  The  apparent 
R.A.  of  the  stars  observed  will  then  be  given  by  the  equations 


On  this  system  we  may  determine  the  R.A.  of  any  number  of 
stars  as  often  as  we  please.  Although,  in  the  present  state  of 
astronomy,  it  is  never  necessary  to  adopt  this  ideal  system,  it  is 
still  true  that  the  latter  embodies  the  fundamental  principles 
on  which  alone  the  absolute  R.A.'s  from  the  equinox  can  be 
determined.  However  complex  the  process  may  be,  the  R.A.'s 
of  each  star  must  ultimately  depend  upon  a  comparison  with  the 


§  171.]   METHOD  OF  DETERMINING  RIGHT  ASCENSIONS        321 

sun,  direct  or  indirect ;  and  the  R.A.  of  the  sun  must  be  regarded 
as  a  function  of  its  observed  declination.  But,  practically,  this 
dependence  of  the  stars  upon  the  sun  is  brought  about,  not 
directly,  but  only  indirectly,  through  correcting  long  series  of 
observations  so  as  to  bring  the  results  of  observation  into 
.accordance. 

171.  Practical  method  of  determining  right  ascensions. 

It  was  pointed  out  in  the  preceding  chapter  that  there  is  no 
immediate  necessity  for  referring  the  R.A.'s  of  the  stars  to  the 
-actual  equinox,  and  that,  except  for  the  necessity  of  comparing 
the  RA.'s  and  longitude,  any  other  origin  would  serve  the 
purpose  as  well  as  the  equinox.  But  there  is  no  visible 
point  or  system  of  points  in  the  sky  which  can  be  used 
to  define  such  an  origin.  The  equinox,  or  as  near  an 
-approximation  to  it  as  can  practically  be  made,  is  therefore 
in  permanent  use. 

A  concession  from  rigour  is,  however,  made  by  regarding  as 
known  quantities  the  R.A.'s  of  a  system  of  fundamental  stars 
-extending  round  the  circle  of  R.A.,  at  not  too  great  a  distance 
from  the  equator,  and  using  them  to  define  the  equinox. 

The  system  now  universally  adopted  is  as  follows  : 

Let  us  put : 

oc13  oc2,  ...ocn,  the  adopted  R.A.'s  of  the  fundamental  stars 
observed  in  the  course  of  any  one  day  or  evening. 

These  R.A.'s  may,  in  all  ordinary  cases,  be  taken  from  one 
of  the  ephemerides.  The  positions  of  the  stars  so  used  are 
practically  more  exact  than  any  single  observation  that  can  be 
made  upon  them. 

Tlt  T2,  T3, ...  Tn,  the  clock  times  of  transit  of  these  stars. 

Then,  as  in  the  case  of  the  sun,  a  correction  of  the  clock  will 
be  derived  for  each  observation  by  subtracting  T  from  a.  The 
several  values  of  the  corresponding  sidereal  times,  commonly 
taken  to  the  tenth  of  an  hour  only,  may  be  arranged  in  a  table 
in  the  form  shown  in  the  following  example,  for  which  the 
numbers  are  derived  from  the  Greenwich  observations  for  1901, 
January  4. 

N.S.A.  X 


322 


THE   POSITIONS   OF   STARS 

Royal  Observatory,  Greenwich,  1901,  Jan.  4. 


[§  171. 


Star 
Observed. 

Clock 
Time. 

T. 
Seconds 
of  Transit. 

a. 
Seconds 
of  R.A. 

A6'obs. 

AC-CUD. 

<*ob§. 

Aa. 

a  Aquila 

h. 
19-8 

59S;80 

s. 

56-77 

-3-03 

-31)3 

56*77 

os-oo 

a  Aquarii 

22-0 

44-95 

41-98 

2-97 

3-05 

41-90 

-0-08 

t   Pegasi 

22-0 

27-03 

24-09 

2-94 

3-05 

23-98 

-0-11 

f  Pegasi 

22-6 

34-75 

31-66 

3-09 

3-05 

31-70 

+  0-04 

M  Pegasi 

22-8 

16-76 

13-69 

3-07 

3-05 

13-71 

+  0-02 

K  Piscium 

23-4 

54-86 

51-87 

2-99 

3-06 

51-80 

-0-07 

t   Piscium 

23-6 

55-02 

51-97 

3-05 

3-06 

51-96 

-o-oi 

7   Pegasi 

o-i 

12-05 

8-99 

3-06 

3-06 

8-99 

o-oo- 

jS  Arietis 

1-8 

14-72 

11-64 

3-08 

3-07 

11-65 

+0-01 

a  Arietis 

2-0 

40-12 

37-04 

3-08 

3-08 

37-04 

o-oo- 

<T  Arietis 

2-8 

6-42 

3-25 

3-17 

3-08 

3-34 

+  0-09 

a  Ceti 

3-0 

10-96 

7-88 

3-08 

3-08 

7-88 

o-oa 

5  Arietis 

3-1 

3-09 

59-89 

3-20 

3-08 

o-oi 

+  0-12 

e  Eridani 

3-5 

20-71 

17-63 

-3-08 

-3-09 

17-62 

-o-oi 
o-ooo 

Mean 

0-3 

-3-06 

The  column  following  the  name  of  the  star  gives  the  clock 
time  of  transit  over  the  meridian  to  the  nearest  tenth  of  an 
hour.  This  is  practically  the  same  as  the  hour  and  tenth  of 
the  star's  R.A. 

Column  T  gives  the  seconds  and  fractions  of  a  second  of  clock 
time  of  transit  as  derived  from  the  observation. 

Column  a  gives  the  seconds  of  .computed  KA.  of  the  star  as 
determined  from  the  ephemeris,  applying  such  small  corrections 
as  were  deemed  necessary. 

The  differences  of  these  two  numbers  is  the  clock  correction 
as  derived  from  each  separate  star. 

The  mean  of  all  the  times  in  the  second  column  is  then  taken 
and  the  means  of  all  the  observed  clock  corrections.  It  is 
assumed  that  the  mean  correction,  —3'06  s.,  is  the  true  correction 
of  the  clock  at  the  mean  of  the  clock  times,  or  0'3  h.  sid.  time. 

By  a  comparison  of  the  observations  on  the  preceding  and 
following  days  it  was  found  that  the  daily  rate  of  the  clock 


§  171.]   METHOD  OF  DETERMINING  RIGHT  ASCENSIONS        323 

was  —0*18  s. ;  with  this  correction  and  rate  the  concluded 
clock  correction  is  found  for  each  observation  of  the  series  by 
the  equation  AC=  A<7m-0-18*  s. 

The  values  of  the  clock  correction  thus  computed  are  found  in 
the  column  following  those  observed.  By  applying  them  to  the 
several  clock  times  of  transits,  the  R.A.  of  each  star  as  derived 
from  the  observation  is  found,  and  its  seconds  given  in  the 
column  ocobs.- 

The  column  Aa  gives  the  correction  to  the  adopted  R.A.  of 
each  star  as  inferred  from  these  observations.  The  concluded 
clock  corrections  are  also  applied  to  the  times  of  transit  of  all 
the  other  stars,  planets,  and  other  bodies  which  may  have  been 
observed,  and  thus  their  apparent  R.A.'s  are  derived. 

At  present,  however,  we  are  concerned  only  with  the  observa- 
tions of  the  fundamental  stars,  and  especially  with  the  nature 
of  the  corrections  derived  in  the  preceding  way.  The  main 
point  is  that  the  R.A.'s  derived  from  the  observations  are  not 
completely  independent  determinations,  because  that  of  each 
star  is  derived  from  the  assumed  R.A.'s  of  all  the  stars  observed, 
itself  included.  It  is  evident  that,  on  this  system,  the  mean 
value  of  all  the  corrections  Aoc  will  vanish,  or  the  mean  of  all 
the  R.A.'s  of  a  group  of  stars  observed  on  any  one  day  will  come 
out  the  same  as  the  mean  of  the  adopted  R.A.'s,  some  being 
increased  and  others  diminished,  so  as  to  bring  the  whole  in 
agreement  among  themselves.  Hence,  if  the  entire  group  is 
affected  by  any  common  error  Aoc,  the  results  of  the  observations 
will  all  be  affected  by  this  same  error.  In  summing  up  the 
results  for  a  year  or  a  series  of  years,  all  the  R.A.'s  derived  from 
observation,  whether  on  the  fundamental  or  other  stars,  will 
therefore  be  affected  by  a  series  of  small  errors 

AOC.J,  Aoc2,  Aot3, . . . , 

which  will  be  the  mean  errors  of  all  the  adopted  R.A.'s  of  the 
individual  groups  of  stars  used  in  forming  the  clock  correction 
during  the  entire  period. 

Entrance  of  systematic  errors.  In  the  case  of  any  individual 
star  the  final  error  will  be  the  mean  of  the  errors  of  all  the 


324        .-  '  THE   POSITIONS   OF  STARS  [§  171. 

stars  with  which  it  was  compared,  these  errors  being  weighted 
on  a  system  to  be  explained  presently.  If  all  the  errors  in  the 
adopted  R.A.'s  of  the  individual  stars  could  be  regarded  as 
independent  and  accidental  ones,  each  as  likely  to  be  positive  as 
negative,  the  final  results  would  be  free  from,  systematic  error. 
But,  as  a  matter  of  fact,  the  adopted  R.A.'s  may  be  affected  by 
systematic  errors  of  two  kinds  :  one  constant,  the  other  varying 
with  the  E.A.  It  is  evident  that  any  systematic  error  in  the 
observations  of  the  sun  may  result  in  the  adopted  R.A.'s  of 
the  stars  being  measured  from  a  point  slightly  different  from 
the  actual  equinox.  Such  a  displacement  of  the  equinox  will 
result  in  all  the  R.A.'s  being  in  error  by  a  quantity  equal  to 
that  displacement.  As  already  pointed  out,  this  constant  error 
is  not  of  Serious  import  unless  in  exceptional  cases  where  ecliptic 
longitudes  have  to  be  used. 

172.  Elimination  of  systematic  errors. 

It  is,  however,  of  the  first  importance  to  eliminate  systematic 
errors  varying  with  the  R.A.  A  study  of  the  conditions  of 
observation  show  how  errors  of  this  class  may  be  perpetuated. 
Let  S  be  any  individual  star  and  Sf  any  fundamental  star  which 
has  been  used  in  deriving  the  clock  error  from  which  S  is 
determined.  Let  N  be  the  number  of  stars  used  on  any  one 
evening  in  determining  these  clock  corrections,  and  let  A/  be  the 
error  in  the  adopted  R.A.  of  Sf.  The  R.A.  of  S  resulting  from 
the  observation  will  in  consequence  of  this  error  A/  be  affected 
by  the  error 


Taking  for  Sf  all  the  fundamental  stars  which  have  been  used 
in  determining  S,  we  see  that  the  latter  will  be  affected  by  a 
certain  mean  error  of  all  the  fundamental  R.A.'s  used  in  de- 
termining the  clock  correction  for  S,  these  means  being  weighted 
in  proportion  to  the  number  of  times  that  each  star  Sf  was  used 
for  the  clock  correction.  Now  if,  in  the  case  of  the  star  S,  the 
stars  8f  were  equally  scattered  all  around  the  circle  of  R.A.,  the 
result  for  tf  would  be  affected  only  by'the  constant  error  common 
to  all.  But,  as  a  matter  of  fact,  observations  are  not  ordinarily 
extended  through  more  than  a  few  hours  of  any  one  night,  and, 


§  173.]  ELIMINATION  OF  SYSTEMATIC  ERRORS  325 

occasionally,  a  longer  period  during  the  day.  The  result  will  be 
that  the  error  of  S  will  not  be  the  mean  error  of  all  the 
fundamental  stars,  but  mainly  of  those  which  culminated  within 
a  short  interval,  say  2,  3,  or  4  hours,  of  8.  It  follows  that,  if 
the  values  of  Aoc  are  systematically  different  in  different  hours 
of  the  circle  of  R.A.,  this  error  will  be  perpetuated  with  only  a 
greater  or  less  diminution.  As  the  adopted  R.A.'s  are  corrected 
from  observations  from  time  to  time,  the  tendency  will  be  to 
smooth  off  the  systematic  errors  in  question  so  that  they  shall 
approximate  to  the  form 

Aoc  =  a  cos  R. A.  +  b  sin  R. A. 

That  is  to  say,  there  will  be  a  periodic  error  in  the  R.A.'s  of 
all  the  stars  which  can  be  eliminated  only  by  comparing  the 
clock  errors  derived  from  stars  as  far  apart  as  possible  in  R.A. 
This  periodic  error  will  be  considered  in  a  subsequent  section. 
At  present  we  pass  to  the  constant  error  of  the  equinox,  which 
we  call  E,  the  equinoxial  error. 

173.  Reference  to  the  sun — the  equinoxial  error. 

We  recall  that  the  determination  of  this  error  must  rest 
fundamentally  upon  observations  of  the  sun.  Ideally  we  have 
considered  the  R.A.  of  the  sun  as  determined  for  each  day. 
Practically,  however,  such  a  determination  need  not  be  made. 
The  practical  method  consists  in  determining  the  error  of  the 
sun's  tabular  R.A.  as  found  for  every  day  of  the  year  in  the 
epherneris,  by  systematic  observations  of  the  transit  of  the  sun 
over  the  meridian,  through  considerable  periods  of  time.  What 
we  then  have  to  deal  with  is  not  the  R.A.'s  and  Decs,  of  the  sun 
as  derived  directly  from  observations,  but  small  corrections  to 
the  values  of  these  quantities  as  tabulated  in  the  Ephemeris. 

The  steps  of  the  process  are  as  follows : 

1.  The  R.A.  and  Dec.  of  the  sun  are  observed  on  as  many  days 
as  possible  through  the  whole  of  one  or  more  years,  and  the 
R.A.'s  are  reduced  as  if  the  sun  were  a  star ;  that  is,  the  clock 
correction  used  is  that  derived  from  the  adopted  R.A.'s  of  funda- 
mental stars.  All  the  observed  R.A.'s  of  the  sun  will,  therefore, 
be  affected  by  the  same  equinoxial  error  as  those  of  the  stars. 


326  THE  POSITIONS  OF   STARS  [§  173. 

2.  Each  E.A.  and  Dec.  of  the  sun  derived  from  the  observations 
is  compared  with  the  positions  of  the  sun  given  in  the  Ephe- 
meris,  and  the  difference  taken.  Leaving  out  accidental  errors 
of  observation,  the  residual  differences  between  the  observed  and 
tabular  positions  are  conceived  to  be  due  to  three  causes  : 

1.  The  equinoxial  error. 

2.  A  constant  error  in  measuring  all  the  declinations  of  the 
sun,  which  may  arise  from  various  causes. 

3.  The  error  of  the  obliquity  of  the  ecliptic  adopted  in  the 
ephemerides. 

To  show  how  these  errors  are  determined,  let  us  put : 

X,  ot,  S,  the  longitude,  R.A.,  and  Dec.  of  the  sun  at  any  moment ; 

€,  the  obliquity  of  the  ecliptic ; 

Et  the  equinoxial  correction. 

The  declinations  of  the  sun  as  given  in  the  ephemeris  are 
derived  from  the  values  of  its  longitude  computed  from  tables  of 
the  sun's  motion.  From  these  longitudes  the  declinations  are 
computed  by  formulae  equivalent  to 

sin  S  =  sin  e  sin  X. 
We  have  also  the  relations 

cos  X  =  cos  oc  cos  S, 
cos  e  sin  X  =  sin  a  cos  S. 

From  these  equations  we  form  equations  of  condition  by  the 
method  set  forth  in  the  chapter  on  Least  Squares.  By  differen- 
tiating the  first  equation  and  substituting  the  others  we  find 

dS  =  cos  oc  sin  e  d\  +  sin  a  de. 
If  we  put 

AX,  the  correction  to  the  longitude  on  any  one  day ; 

Ae,  the  correction  to  the  obliquity  of  the  ecliptic,  which  may 

be  regarded  as  constant  through  the  entire  period ; 
A0,  a  possible  constant  error  in  measuring  all  the  declinations 

with  the  instrument ; 

A£,  the  excess  of  the  observed  over  the  tabular  declination; 
then  each  observation  of  declination  will  give  the  equation  of 
condition  :  cog  a  gin  g  Ax  +  sin  a  Ae+ An  =  A& 


§173.]  REFERENCE  TO  THE   SUN  327 

If  the  tabular  elements  of  the  earth's  orbit  around  the  sun  are 
•correct,  AX  is  constant  throughout  the  entire  period  of  observation. 
In  all  probability  the  errors  of  the  elements  are  so  small  that  we 
may  regard  their  possible  effect  upon  the  result  as  quite  in- 
significant. Assuming  AX  as  a  constant,  we  have  to  solve  the 
^bove  equations  of  condition  by  the  method  of  least  squares. 

It  is  not,  however,  necessary  to  treat  the  corrections  in- 
dividually. The  values  of  the  coefficients  sin  a.  and  cos  a  vary 
so  slowly  and  regularly  that  we  may  use  their  mean  values 
for  each  month  as  constant  throughout  the  month.  We  then 
have  twelve  equations,  one  derived  from  the  observations  of  each 
month.  We  may  assign  to  these  equations  weights  proportional 
to  the  number  of  observations.  But,  unless  the  latter  are  very 
unequally  divided  through  the  year  (a  circumstance  which  will 
greatly  impair  their  value,  and  perhaps  render  them  scarcely 
worth  using),  we  shall  get  as  good  a  result  by  assigning  equal 
weights  to  the  observations  of  each  month  as  if  we  assigned 
weights  dependent  upon  the  number  of  observations.  In  fact, 
the  errors  which  we  have  to  fear  are  not  the  purely  accidental 
errors,  but  possible  constant  errors  continuing  through  a  month, 
but  varying  from  one  month  to  another.  Their  possibility  will 
lead  us  to  diminish  the  weight  assigned  to  a  large  number  of 
observations  in  a  single  month,  so  as  to  make  it  approximate  to 
the  weights  assigned  to  other  months.  As  a  general  rule,  the 
mean  of  the  dates  of  observations  in  each  month  will  not,  in 
the  general  average,  be  greatly  different  from  the  middle  of  the 
month.  We  may,  therefore,  conceive  the  twelve  monthly  values 
of  oc  to  form  a  series  scattered  at  equal  intervals  of  30°  each 
around  the  circle.  Thus  we  shall  have  twelve  conditional 
equations : 

cos  04  sin  e  AX  +  sin  04  Ae  +  A0  =  A^, 
cos  oc2  sin  e  AX  +  sin  oc2  Ae  +  A0  =  A£2, 
cos  oc3  sin  e  AX  +  sin  oc3  Ae  +  A0  =  A<53, 

cos  oc12  sin  e  AX  +  sin  oc12  Ae + A<j  =  A<512. 

Proceeding  now  to  the  method  of  solution  by  least  squares,  the 
normal  equation  in  AX  may  be  found  by  multiplying  each  equation 


328  THE  POSITIONS  OF   STARS  [§  173_ 

by  the  coefficient  cos  oc.  It  is  true  that  the  actual  value  of  the 
coefficient  of  AX  is  cos  oc  sin  e,  and  if  we  multiply  by  this  coefficient 
all  our  products  will  contain  the  common  factor  sin2e.  But  it 
will  be  more  convenient  to  regard  sine  AX  as  the  unknown 
quantity,  which  we  may  call  x.  The  general  form  of  the  equations- 
will  then  be 


oC;Ae+A0  =  A4 
where  i  =  l,2,  3,...  12. 

The  normal  equation  in  x  derived  by  the  method  of  §  36  wills 
now  be 

2  cos2oc  .  x  +  2  sin  oc  cos  oc  .  Ae  +  2  cos  oc  .  A0  =  2  cos  a  A& 

When  the  values  of  oc  are  scattered  equally  around  the  circle., 
we  have  by  known  trigonometric  theorems, 

2cos2oc  =  6, 
2  sin  oc  cos  oc  =  0, 

2  cos  oc  =  0. 

Thus  our  normal  equation  in  x  reduces  to  the  very  simple  forms 

6x  =  2  cos  oc  A<S, 
and  the  correction  AX  is  given  in  the  form 

A  .        x       2  cosoc  AS 
AX  =  —.  —  =  —  ^-77.  —  . 
sine          2'40 

Having  thus  found  AX,  we  have  next  to  determine  its  effect 
upon  the  R.A.'s.  In  the  case  of  the  sun,  this  coordinate,  as- 
tabulated  in  the  ephemeris,  is  derived  from  the  equation 

tan  oc  =  cos  c  tan  X. 
We  have,  by  differentiation, 

sec2oc  doi  =  cos  e  sec2X  d\  —  sin  e  tan  X  de. 
We  have  also  cos  a  cos  8  =  cos  X, 

wh  ence  da.  =  cos  e  sec2<5  d\  —  cos  a  tan  8  de. 

The  mean  value  of  cos  e  sec2c5  in  the  course  of  the  year  may  be 
regarded  as  1,  and  that  of  cos  octant  as  0.  We  may  therefore 
put,  as  the  mean  result  of  the  entire  series  of  observations  of  the- 

sun>  = 


§  173.]  CORRECTION  OF  THE  EQUINOX  32£ 

which  we  regard  as  the  definitive  correction  to  the  tabular  R.A. 
of  the  sun. 

We  have  also  found  a  series  of  apparent  corrections  to  the 
sun's  tabular  R.A.  by  the  reduction  of  the  observations  in  R.A. 
Let  us  put,  for  any  day, 

oc  comp.,  the  tabular  R.A.  of  the  sun  ; 

oc  obs.,  the  R.A.  derived  from  the  observed  times  of  transit,  by 
applying  the  method  of  §  171,  using  the  adopted  R.A.  of 
the  fundamental  stars. 

Then,  we  put  ^  =  a  obs  _  a  CQmp 

All  these  observed  R.A.'s  require  the  common  correction  E. 
Hence  the  actual  correction  to  the  tabular  R.A.  of  the  sun  is 


Equating   this  to  the  actual  value  of  AocG  found  from   the- 
observed  declinations,  we  have 


in  which  we  may  use  for  A'ot0  its  mean  value  for  the  entire  series 
of  observations. 

It  may  seem  that  the  number  of  quantities  which  we  have 
had  to  change  or  drop  in  order  to  reduce  our  result  to  this 
simple  form  is  so  great  that  the  errors  thus  arising  may  be 
important.  But  this  will  be  the  case  only  when  the  observations 
are  very  unequally  scattered  throughout  the  year.  If  there  are 
an  equal  number  of  observations  in  every  month,  the  normal 
equation  will  be  found  to  reduce  to  this  form  ;  that  is  to  say,  the 
corrections  Ae  and  A<5  will  be  eliminated  of  themselves.  If  the 
observations  are  scattered  very  unevenly  through  the  year, 
especially  if  comparatively  few  are  made  in  some  months,  the 
equations  of  condition  must  be  solved  by  assigning  weights  to 
the  mean  result  for  each  month.  The  normal  equation  will  then 
no  longer  reduce  to  the  above  simple  form,  and  must  be  solved 
in  the  regular  way. 

The  determination  of  the  correction  as  here  set  forth  com- 
prises two  steps,  one  the  determination  of  the  correction  to  the- 


330  THE   POSITIONS  OF  STARS  [§  173. 

sun's  absolute  R.A.  from  the  observations  of  its  declination,  the 
other  that  of  the  differences  between  the  E.A.'s  of  the  stars 
and  of  the  sun.  These  two  determinations  may  be  considered 
as  quite  independent  of  each  other.  The  equinox  can  be  deter- 
mined from  observations  of  the  declination  alone  made  at  some 
observatories  and  of  R.A.'s  alone  at  other  observatories.  Now 
that  the  change  in  the  error  of  the  sun's  longitude  in  the  course 
of  a  year  is  so  small  as  to  be  completely  masked  in  the  errors  of 
the  observations,  this  method  of  independent  determination  of 
the  two  quantities  is  the  better  one  to  adopt. 

Another  consideration  bearing  on  the  case  is  that  the  personal 
equation  of  the  observers  in  observations  of  the  sun's  limb  is 
probably  different  from  that  for  observations  of  the  stars,  and  that 
this  difference  is  far  from  being  the  same  with  different  observers. 

174.  The  general  policy  in  the  construction  of  the  national 
Ephemerides  has  recently  been,  and  still  is,  not  to  change  the 
adopted  equinox  until  a  long  series  of  observations  at  different 
observatories  shall  show  a  well-marked  and  undoubted  correction 
to  be  necessary.  The  equinoxes  now  in  use,  that  is  to  say,  the 
mean  R.A.'s  of  the  fundamental  stars,  were  determined  in  1876 
from  all  the  best  observations  then  available.  The  general  mean 
result  of  recent  observations  seems  to  indicate  a  positive  correc- 
tion to  the  R.A.  of  all  the  stars ;  but,  from  the  very  nature  of  the 
case  the  results  are  somewhat  discordant,  and  the  amount  of  the 
correction  is  still  doubtful.  Its  reality  is  yet  more  questionable 
in  the  light  of  the  recently  recognized  "magnitude-equation" 
now  known  to  affect  the  R.A.'s  of  all  the  stars  observed  up  to 
the  present  time.  The  existence  of  this  equation  renders  it 
probable  that  the  general  R.A.'s  of  the  stars  determined  in  past 
times  may  have  been  too  small  to  an  extent  to  more  than 
neutralize  the  possible  positive  correction  to  the  R.A.'s  of  all 
the  stars,  which  we  have  mentioned  as  indicated  by  recent 
observations. 

175.  The  Greenwich  method. 

The  only  observatory  which,  at  the  present  time,  makes  it  a 
point   to   determine   the   equinoxes    every   year   from   its   own 


§  176.]  CORRECTION   OF  THE   EQUINOX  331 

observations  is  that  of  Greenwich.  Here  a  method  devised  by 
Airy  is  used,  which,  though  involving  the  general  principles  just 
set  forth,  deviates  in  detail. 

The  division  of  the  equinoxial  correction  into  two  parts,  the 
one  applicable  to  the  sun's  tabular  R.A.,  the  other  to  the  differ- 
ences between  the  R.A.'s  of  the  sun  and  stars,  is  not  recognized. 
The  method  consists  in  taking  the  apparent  corrections  to  the 
sun's  R.A.  and  Dec.  obtained  in  the  usual  way,  and  converting 
them  into  errors  of  the  tabular  longitude  and  latitude.  The 
-combined  effect  of  the  two  errors  Aoc0,  and  A'a0,  is  to  produce  in 
the  errors  of  latitude  an  annual  period  of  the  form 

A/3  =  E  cosec  e  cos  O. 

The  error  Ae  of  the  obliquity  produces  in  A/3  a  term  of  the 
form 


while  there  may  be  a  constant  error  in  all  the  measures  of 
decimation  made  with  the  instrument  in  the  course  of  the  year. 

The  mean  of  all  the  observed  errors  of  the  latitude  during 
each  month,  gives  an  equation  of  the  form 

a  +  b  cos  0  +  c  sin  O  =  A/3, 

and  the  solution  of  all  the  equations  thus  formed  gives  a,  b,  and  c. 
Then,  J£  =  6cosece,  Ae  =  c.  The  result  thus  reached  is  doubtless 
the  same  as  if  the  method  of  the  present  chapter  were  applied. 
But  the  method  does  not  separate  the  two  parts  of  which  the 
correction  E  is  composed. 

Section  II.    The  Determination  of  Declinations. 

176.  The  ideal  meridian  circle. 

The  development  of  the  principles  on  which  declinations  of 
stars  are  determined  requires  a  statement  of  the  fundamental 
idea  of  a  meridian  circle.  The  essential  parts  of  this  instrument 
are  a  finely  graduated  circle  rigidly  attached  to  a  telescope  (see 
Fig.  36).  The  latter  is,  in  principle,  the  transit  instrument  already 
described,  and  therefore  has  but  one  motion,  that  in  the  plane  of 
the  meridian  on  a  fixed  horizontal  east  and  west  axis.  The 


332 


THE   POSITIONS  OF   STARS 


[§  176, 


plane  of  the  circle  is  parallel  to  the  tube  of  the  telescope,  and  in 
the  plane  of  the  meridian.  At  the  eye  end,  in  the  focus,  is  a 
horizontal  spider  line,  at  right  angles  to  the  vertical  line  over 
which  transits  are  observed.  The  result  of  these  rigid  con- 
nections is,  that  when  the  instrument  is  turned  on  its  axis  the 
angular  motion  of  the  line  of  sight  is  equal  to  the  angle 
through  which  the  circle  has  turned.  We  have  to  show  how 
this  angle  is  measured. 


FIG. 


The  circumference  of  the  circle  is  divided  into  360  parts 
of  one  degree  each  by  fine  lines  or  graduations,  and  each  of 
these  is  subdivided  into  equal  parts,  generally  2'  or  5'  each.  The 
graduations  are  numbered  from  0°  to  360°,  and  are  visible 
through  at  least  one  pair  of  microscopes  at  opposite  ends  of  a 
diameter.  These  microscopes  are  firmly  fixed  to  the  supporting 
pier,  and  therefore  do  not  revolve  with  the  instrument.  For  our 
present  purpose,  we  need  consider  only  a  single  microscope. 
This  is  supplied  with  a  micrometer,  by  means  of  which  the 
position  of  any  graduation  in  the  field  of  view  of  the  microscope 
may  be  accurately  measured. 

The  result  of  this  combination  is  that  the  varying  position 
of  the  circle  becomes  a  continuous  quantity,  that  is,  motions 
ever  so  small  may  be  measured.  Suppose,  to  fix  the  ideas,  that 
the  graduation  28°  16'  is  exactly  in  a  certain  part  of  the  field  of 
the  microscope,  which  we  take  as  the  zero  point.  Then  we  say 
that  the  circle-reading  is  28°  16'  0"'0.  Then,  give  the  circle  a 
small  motion  forward.  If  we  find  that  the  micrometer,  when 
set  on  the  graduation  28°  16',  now  reads  7"'5,  we  say  that  the 
circle-reading  is  28°  16'  7"'5,  and  we  know  that  the  circle  has 
been  moved  through  an  angle  of  7"'5. 


;§  177.]  THE   IDEAL  MERIDIAN  CIRCLE  333 

It  follows  that,  corresponding  to  each  position  of  the  circle, 
•and  therefore  to  each  direction  of  the  line  of  sight  of  the  telescope, 
there  is  a  certain  circle-reading.  A  fundamental  principle  of  the 
method  is,  that  a  single  reading  tells  us  nothing  of  the  absolute 
direction  of  the  line  of  sight;  but  that  the  difference  between 
two  readings  is  equal  to  the  arc  through  which  the  circle  and 
telescope  have  turned  between  the  readings. 

In  an  instrument  of  perfect  stability  the  circle  reading  should 
.always  remain  the  same  for  the  same  direction  of  the  telescope. 
The  direction  might  be  the  zenith,  the  nadir,  the  equator,  or  the 
pole.  But,  as  a  matter  of  fact,  the  reading  for  any  fixed  point 
changes  by  minute  amounts  not  only  from  day  to  day,  but  even 
through  different  hours  of  the  day.  The  determination  of  these 
changes  forms  one  of  the  most  troublesome  problems  with  which 
the  observer  has  to  deal.  We  shall  begin  by  ignoring  them,  and 
showing  how  positions  of  the  stars  are  determined,  supposing  the 
instrument  stable. 

177.  Principles  of  measurement. 

Supposing  our  instrument  ideally  perfect,  which,  in  practice, 
it  never  is,  we  have  to  show  how  fundamental  declinations  are 
measured  with  it.  In  doing  this  it  will  be  convenient  to  replace 
the  declination  by  the  polar  distance,  from  which  we  can,  at  any 
time,  pass  to  declination  by  the  simple  process  of  subtraction 
from  90°.  The  polar  distance  of  a  star  being  defined  as  its 
angular  distance  from  the  North  Pole,  its  determination  would 
be  extremely  simple  if  only  the  pole  were  a  visible  point  in  the 
heavens.  We  should  set  the  instrument  on  the  pole  and  de- 
termine the  circle  reading  =  Cp.  We  then  should  point  the 
instrument  at  a  star  as  it  passes  the  meridian,  and  call  the  circle 
reading  C8.  The  difference,  Cs—Cp,  corrected  for  refraction,  is 
the  polar  distance  of  the  star  as  given  by  the  instrument,  and 
9(r-(Cs-Cp)  is  its  declination. 

The  pole,  not  being  a  visible  point  in  the  heavens,  has  to  be 
otherwise  defined.  Its  true  position,  being  the  line  of  the  in- 
stantaneous axis  of  rotation  of  the  earth,  is  midway  between  the 
points  at  which  a  star  near  the  pole  crosses  the  meridian  at  an 


334  THE   POSITIONS  OF  STAES  [§  177. 

upper  and  lower  culmination.  It  follows  that  if  we  put  Gu  for 
the  circle  reading  when  a  circumpolar  star  crosses  the  meridian 
above  the  pole,  and  Cl  for  the  reading  when  it  passes  below  the 
pole  twelve  hours  later,  we  shall  have 


always  supposing  the  zero-point  constant  during  the  twelve 
hours. 

With  this  value  of  Cp  the  polar  distance  and  declination  of 
any  star  will  be  given  by  the  preceding  equations.  No  funda- 
mental determinations  of  declinations  can  be  made  except  in 
this  way. 

If  the  instrument  were  perfectly  stable,  that  is  to  say,  if  the 
circle  reading,  when  the  telescope  is  set  on  the  pole,  were  the 
same  for  a  whole  year,  the  essential  principles  of  the  method  as 
thus  set  forth  would  be  complete.  But,  as  a  matter  of  fact,  the 
reading  Cp  may  change  from  day  to  day,  or  even  from  hour  to- 
hour.  This  renders  it  necessary  to  have  some  fixed  point  of 
reference,  the  absolute  position  of  which  is  arbitrary,  but  which 
can  be  determined  at  any  time.  Let  TV  be  the  circle  reading  for 
such  a  point,  which  we  call  the  zero-point,  and  which  we  suppose 
to  be  determined  as  often  as  necessary.  Then  making  the 
observations  above  described,  we  have  only  to  substitute  C—N 
for  C  ;  in  other  words,  we  may  subtract  the  reading  N  for  the 
moment  of  observation  from  all  the  observed  circle  readings  on 
the  stars,  and  use  the  difference  instead  of  C. 

More  specifically,  let  us  suppose  that,  at  three  different  times,, 
we  observe 

A  circumpolar  star  at  upper  transit  ; 

The  same  star  at  lower  transit  ; 

Any  other  star,  S. 

As  before,  let  the  three  circle  readings  for  these  stars  be 

Cu,  Cl,  and  C.. 

Also  let  the  circle  readings  for  the  zero-point  at  the  three 
times  be 

X    N»  and  N. 


§  178.]  PRINCIPLES  OF  MEASUEEMENT  335 

We  then  put  C'U  =  CU-NV 

O'^G,-^, 

V.  =  G.-NV 

and  we  shall  have 

Polar  Distance  of  Star  =  (7s-i((7w +£',). 

Through  this  process  results  will  be  the  same  as  if  the 
stability  of  the  instrument  were  perfect. 

For  the  zero-point  in  question  the  nadir  is  now  almost 
universally  used.  It  is  determined  by  pointing  the  telescope 
vertically  downward  at  the  surface  of  a  basin  of  quicksilver, 
and  applying  certain  devices  by  which  the  verticality  of  the 
line  of  sight  may  be  ascertained.  The  details  of  the  method 
belong  to  the  subject  of  instrumental  astronomy,  and  cannot 
be  entered  upon  here.  For  our  present  purpose,  the  nadir  is 
simply  a  fixed  direction  for  which  the  circle  reading  may  be 
determined  at  any  time. 

In  the  preceding  outline  we  have  left  out  of  consideration 
the  various  corrections  due  to  precession,  nutation,  refraction, 
instrumental  errors,  and  other  causes,  in  order  to  facilitate  the 
reader's  grasp  of  the  essential  process.  The  latter  results  in 
independent  determinations  of  the  absolute  declinations  of  the 
stars,  substantially  the  same  as  if  the  pole  were  a  visible  point 
at  which  the  instrument  could  be  pointed  at  any  time.  This 
is  the  ideal  result  at  which  work  with  an  instrument  should  aim. 

178.  Differential  determinations  of  declination. 

We  now  pass  to  differential  determinations.  In  these  the 
polar  point  and  the  fixed  ^-point  are  determined  from  the 
declinations  of  fundamental  stars,  assumed  to  be  known  in 
advance.  Let  us  put 

3,  the  known  or  assumed  declination  of  such  a  star ; 

(78,  the  circle  reading  when  the  instrument  is  set  on  this  star ; 

Ceq,  the  circle  reading  when  the  instrument  is  pointed  at  the 
equator ; 

Cp,  the  reading  for  the  pole. 

Since  the  arc  from  the  star  to  the  equator  is  equal  to  the 
declination  of  the  star,  it  follows  that,  having  made  the  observa- 


336  THE  POSITIONS  OF  STARS  [§  178. 

tion  of  G8y  we  infer  that,  if  the  instrument  were  pointed  at  the 
celestial  equator,  the  circle  reading  would  be 


and  for  the  pole,  Cp=Ct+8-  90°. 

If  C's  is  the  reading  for  any  other  star,  and  we  put  S'  for  its 
declination,  we  shall  have 


In  practice  a  number  of  standard  stars  are  observed  in  the 
•course  of  an  evening,  from  each  of  which  a  value  of  Ceq  is 
-derived.  The  mean  of  these  values  is  the  value  of  Ceq,  with 
which  the  declinations  of  all  the  other  stars  are  determined  by 
the  above  formula. 

179.  Systematic  errors  of  the  method. 

It  will  be  seen  that  this  method  is  analogous  to  that  applied 
in  the  case  of  R.A.'s.  There  is,  however,  an  important  difference 
in  the  nature  of  the  systematic  errors  to  which  the  method  is 
liable  in  the  two  cases.  The  stars  in  any  one  region  of  the 
heavens,  or  in  any  hour  of  R.A.,  culminate  in  the  course  of  a 
year  at  every  hour  of  the  day  in  succession.  Consequently, 
systematic  errors  arising  from  diurnal  changes  of  temperature 
-and  all  other  causes  which  go  through  their  period  in  the  course 
of  a  day  are  in  great  part  eliminated  from  observations  extending 
through  an  entire  year.  In  other  words,  so  far  as  unavoidable 
systematic  errors  of  observations  are  concerned,  all  the  stars  are, 
so  to  speak,  on  the  same  footing. 

But  this  is  not  the  case  with  the  declinations.  Since  any  star 
always  culminates  at  nearly  the  same  altitude,  any  systematic 
error  depending  upon  the  zenith  distance  will  repeat  itself 
indefinitely.  It  is  found  from  experience  that  the  declinations 
of  stars  given  by  different  instruments  show  very  appreciable 
systematic  differences.  In  good  instruments  the  difference  rarely, 
if  ever,  amounts  to  a  second  of  arc,  but  may  be  an  important 
fraction  of  a  second.  In  the  imperfect  instrument  of  former 
times  it  may  have  been  greater  than  a  second. 


§179.]  SYSTEMATIC  ERRORS  OF   DECLINATIONS  337 

The  result  is  that,  if  we  compare  the  equatorial  or  polar  point 
of  the  instrument  derived  from  a  group  of  stars  in  one  declina- 
tion, it  may  be  systematically  different  from  that  derived  from  a 
group  in  a  very  different  declination.  If  both  groups  are  com- 
bined, the  result  will  be  a  heterogeneity  in  the  declinations 
finally  derived,  which  may  seriously  detract  from  their  useful- 
ness. Although  the  practical  methods  of  dealing  with  this 
case  are  not  strictly  germane  to  the  present  work,  it  is 
necessary  to  show  their  general  character,  in  order  to  be  able 
to  deal  in  the  most  intelligent  way  with  the  results  actually 
found  in  published  catalogues  of  stars  prepared  from  the  work 
of  different  observatories.  The  principal  reason  for  using  the 
differential  instead  of  the  absolute  method  in  declinations  is  the 
avoidance  of  the  labour  of  repeated  determinations  of  the  nadir 
point  with  an  instrument  which  is  probably  not  stable  through 
any  one  day.  By  determining  the  equatorial  or  polar  point  from 
fundamental  stars,  this  labour  is  avoided  and  more  observations 
•can  be  made.  Such  systematic  errors  as  may  affect  the  result 
need  be  no  greater  than  those  of  the  fundamental  stars  them- 
selves. With  a  fairly  stable  instrument  it  is  possible  to  arrange 
observations  so  that  the  determination  of  the  nadir  point  will 
not  be  really  necessary  even  if  fundamental  results  are  aimed 
at.  There  are  two  methods  of  doing  this. 

The  first  method  is  to  adopt  as  fundamental  stars  only  stars 
quite  near  the  pole,  say  within  10°  or  12°  of  that  point.  The 
systematic  errors  in  the  declinations  of  stars  are  smaller  the 
nearer  they  are  to  the  pole ;  and  within  the  distance  above 
mentioned  they  may  be  regarded  as  unimportant  for  ordinary 
purposes.  Moreover,  their  effect  will  be  almost  entirely  elimi- 
nated if  the  stars  are  observed  both  above  and  below  the  pole,  as 
they  pass  the  meridian  at  the  upper  and  lower  culmination.  In 
this  way  the  polar  reading  of  the  instrument  may  be  determined 
from  each  night's  work,  and  determinations  of  declination, 
practically  absolute,  may  be  made  at  all  declinations. 

The  same  result  will  be  reached  if  the  adopted  declinations 
are  made  to  agree  with  those  determined  by  the  instrument 
itself  either  by  the  above  method  or  by  the  absolute  method. 
N.S.A.  Y 


338  THE  POSITIONS  OF   STARS  [§  179. 

It  is,  therefore,  not  necessary  to  follow  the  method  rigorously 
year  after  year.  If,  by  adopting  it,  corrections  are  found  for 
a  limited  number  of  fundamental  stars,  and  the  corrected  values 
of  the  latter  are  then  used  for  the  equatorial  or  polar  point r 
it  will  not  be  necessary  to  continue  the  observations  of  the 
polar  stars. 

The  second  method  consists  in  using  as  fundamental  stars  only 
those  included  in  some  one  zone  of  declination,  5°  or  10°  in 
breadth.  The  systematic  differences  within  such  a  zone  may 
probably  be  regarded  as  evanescent.  If  the  stars  to  be  deter- 
mined also  lie  within  the  same  zone,  we  shall  have  a  set  of 
declinations  affected  only  by  the  common  error  of  the  funda- 
mental declinations  in  the  zone,  which  error  we  conceive  to  be 
capable  of  ulterior  determination  and  call  h. 

If,  following  this  method,  stars  scattered  at  widely  different 
declinations  in  the  sky  are  observed,  all  the  resulting  declina- 
tions observed  above  the  pole  will  be  in  error  by  h,  and  those 
below  the  pole  by  —h.  There  will  therefore  be  a  systematic 
difference  of  2h  between  the  declinations  of  the  same  star  when 
observed  above  and  below  the  pole,  which  will  show  the  value 
of  h.  This  being  determined,  all  the  declinations  can  be  cor- 
rected by  this  amount,  and  thus  reduced  to  the  values  which 
the  instrument  itself  would  have  given  had  it  been  used  as 
a  fundamental  one. 

When  neither  of  these  methods  is  applied,  and  when  stars  of 
different  declinations  have  been  used  without  any  discussion  of 
the  systematic  discordances  between  the  results  derived  from 
them,  the  results  cannot  be  regarded  as  fundamental.  Nor  can 
their  accuracy  be  estimated  except  by  a  comparison  with  other 
authorities. 


CHAPTER  XIII. 

METHODS  OF  DERIVING  THE  POSITIONS  AND  PROPER 
MOTIONS  OF  THE  STARS  FROM  PUBLISHED  RESULTS 
OF  OBSERVATION. 

Section  I.    Historical  Review. 

180.  The  Greenwich  Observations. 

The  material  at  the  command  of  the  astronomer  for  the 
determination  of  positions  of  the  stars  consists  mainly  in 
catalogues  of  such  positions  at  various  epochs,  as  derived  from 
observations  of  right  ascension  and  declination  made  at 
observatories  during  the  past  two  centuries.  Owing  to  the 
diversity  in  the  construction  of  the  instruments  of  observation, 
in  the  method  of  using  them,  and  in  the  adopted  systems  of 
deriving  and  publishing  their  results,  the  material  in  question 
is  so  heterogeneous  that  few  features  are  common  to  all  the 
catalogues.  The  method  of  utilizing  it  therefore  requires  a 
special  study  of  each  individual  catalogue,  as  well  as  a  com- 
prehensive idea  of  the  nature  of  the  observations  on  which  all 
the  results  depend.  The  mastery  of  the  subject  will  be  facilitated 
by  beginning  with  a  brief  outline  of  the  labours  which  modern 
astronomers  have  undertaken  for  the  purpose  in  question. 

Notwithstanding  the  imperfections  of  his  instruments,  the 
catalogue  of  stars  constructed  by  Flamsteed,  first  Astronomer 
Royal,  during  the  few  years  preceding  his  death,  which  occurred 
in  1719,  was  a  great  improvement  on  any  preceding  work  of  the 


340  OBSERVATIONS   OF  THE   STARS  [§  180. 

kind,  forming  in  a  certain  sense  the  basis  of  Bradley's  work  half 
a  century  later,  as  well  as  determining  its  direction.  The  most 
familiar  remnant  of  Flamsteed's  work  is  the  system  of  numbers 
attached  to  his  catalogue  of  stars,  which  are  still  used  to 
designate  such  of  the  stars  catalogued  by  him  as  are  not  desig- 
nated by  a  letter  on  Bayer's  system.  But  observations  of  the 
accuracy  necessary  to  fixing  the  proper  motions  of  the  stars 
began  with  Bradley,  Astronomer  Royal  of  England,  in  the  middle 
of  the  eighteenth  century.  Previous  to  his  time,  1750  to  1756, 
the  instruments  and  methods  of  determination  were  so  imperfect 
that  it  is  now  possible,  from  the  data  which  have  since  accumu- 
lated, to  compute  positions  of  the  stars  at  any  previous  epoch 
with  a  higher  degree  of  accuracy  than  the  astronomers  of  the 
time  were  able  to  observe  them.  Notwithstanding  the  excellence 
of  Bradley's  observations,  his  instrument  for  measuring  declina- 
tions was  of  the  older  kind. 

It  was  not  till  half  a  century  after  his  time  that  the  advantages 
of  using  a  complete  circle,  graduated  through  its  entire  circum- 
ference, for  the  purpose  of  measuring  declinations  was  fully 
understood  by  astronomers.  Down  to  nearly  the  beginning  of 
the  nineteenth  century  the  mural  quadrant  was  the  principal 
instrument  for  this  purpose.  As  implied  by  its  name,  this 
instrument  consisted  of  a  quadrant,  the  actual  arc  of  which  was, 
however,  somewhat  more  than  90°,  attached  to  the  face  of  a  wall 
in  the  plane  of  the  meridian.  The  telescope  moved  on  a  centre 
coincident  with  that  of  the  graduated  arc.  It  is  readily  seen 
that,  how  great  soever  might  be  the  care  and  skill  employed  in 
the  construction,  the  readings  of  the  arc  were  subject  to  errors 
arising  from  the  unavoidable  non-coincidence  of  the  centre  of 
motion  of  the  telescope  with  the  geometric  centre  of  the  quadrant 
— to  errors  of  graduation — and  to  changes  in  the  position  of 
the  instrument  from  time  to  time,  due  to  slow  motions  of  the 
supporting  wall.  Moreover,  observations  during  any  one  period 
could  be  made  only  on  one  side  of  the  zenith.  It  was  therefore 
necessary,  when  observations  were  to  be  made  on  the  other  side, 
to  take  the  quadrant  down  and  remount  it. 

The  telescope  of  the  quadrant  being  supposed  to  move  in  the 


§  180.]  THE   GREENWICH   OBSERVATIONS  341 

plane  of  the  meridian,  sometimes  served  the  purpose  of  a  transit 
instrument  for  the  observation  of  right  ascensions.  The  quadrant 
with  its  telescope  thus  served  in  a  rude  way  the  purpose  of  the 
modern  meridian  circle.  The  use  of  a  separate  transit  instru- 
ment made  its  way  very  slowly,  not  being  introduced  at  the 
Paris  observatory  until  the  beginning  of  the  nineteenth  century. 
Bradley's  observations  in  right  ascensions  were  made  with  this 
instrument,  a  circumstance  to  which  their  superiority  is  largely 
due. 

Bradley's  instruments,  improved  though  they  were,  continued 
to  be  used  at  Greenwich  until  1812-16,  when  the  mural  quadrant 
was  replaced  by  a  mural  circle.  The  work  with  this  instrument 
by  Pond,  Astronomer  Royal  1812-1835,  was  far  superior  to  any 
that  had  preceded  it.  He  added  a  second  mural  circle,  and  made 
observations  with  the  two  conjointly,  so  as  to  obtain  the 
supposed  benefits  of  a  combination.  His  observations  with 
these  circles  have  in  recent  times  been  partly  reduced  by 
S.  C.  Chandler,  who  found  them  to  be  of  a  degree  of  excellence, 
especially  as  regards  their  freedom  from  systematic  errors,  that 
has  rarely  been  exceeded  since  his  time.  It  is  therefore  to  be 
regretted  that,  with  the  exception  of  a  few  fundamental  stars 
discussed  by  Chandler,  no  results  of  Pond's  work  on  the  stars  are 
yet  available  except  those  computed  by  himself,  and  therefore 
derived  by  the  imperfect  methods  then  in  use. 

Pond's  successor  in  the  office  of  Astronomer  Royal  was  George 
Biddle  Airy,  who  held  that  position  from  1836  to  1881,  a  period 
of  forty-five  years.  Airy's  abilities  as  a  planner  and  ad- 
ministrator of  work  were  of  the  highest  order.  His  system  was 
based  on  the  idea  that  one  directing  head  could  work  out  all  the 
formulae  and  prepare  all  the  instructions  required  to  keep  a 
large  body  of  observers  and  computers  employed  in  making  and 
reducing  astronomical  observations.  A  few  able  lieutenants, 
who  would  see  that  all  the  details  were  properly  carried  out, 
were  an  adjunct  of  his  system.  Acting  on  these  ideas,  he 
reduced  the  work  of  the  observatory  to  a  system  more  com- 
prehensive in  its  details  than  anyone  had  ever  before  attempted 
in  the  conduct  of  astronomical  operations.  The  main  object  he 


342  OBSERVATIONS  OF  THE   STARS  [§  180. 

had  in  view  was  the  determining  of  positions  of  the  heavenly 
bodies.  He  adopted  the  system  of  collecting  the  results  of  the 
observations  of  the  stars  from  time  to  time  in  catalogues,  each 
embracing  several  years'  work,  generally  between  six  and  ten. 
The  transit  instrument  and  mural  circle,  both  excellent  of  their 
kind,  were  continued  in  use  until  1850,  when  the  great  transit 
circle,  devised  in  all  its  details  by  Airy,  was  installed,  and  still 
remains  the  principal  meridian  instrument  of  the  observatory. 

This  instrument  has  proved  one  of  the  most  useful  in  all  star 
determinations  except  those  special  ones  demanding  the  highest 
degree  of  delicacy.  Its  construction  is  an  interesting  reflection 
of  Airy's  methods.  He  thought  out  what  he  might  well  suppose 
to  be  all  possible  sources  of  systematic  and  accidental  error, 
devised  means  of  avoiding  or  eliminating  them,  established  a 
complete  system  of  supervision,  and  then  assumed  that  the 
results  of  his  system  could  be  regarded  as  absolute  determina- 
tions, which  needed  little  further  investigation  so  far  as  possible 
corrections  to  their  results  were  concerned.  Acting  on  this 
system,  his  transit  circle  was  not  reversible,  a  quality  necessary 
only  when  it  is  assumed  that  an  instrument  may  give  different 
results  if  the  pivots  are  turned  end  for  end  in  their  bearings. 
The  circle  is  not  adjustable  on  the  axis,  because  the  errors  of 
graduation  were  determined  once  for  all,  and  were  not  supposed 
subject  to  any  further  correction.  No  allowance  was  made  for 
possible  systematic  errors  in  determining  the  line  of  collimation 
of  the  telescope.  In  a  word,  the  idea  that  the  instrument  might 
well  be  subject  to  errors  arising  from  obscure  or  unknown  causes, 
and  that  it  was  desirable  to  vary  in  every  possible  way  the 
methods  of  using  it  in  order  to  test  the  existence  of  such  errors, 
was  not  so  prominent  in  his  mind  as  it  was  in  that  of  the  school 
of  Bessel. 

One  result  of  this  is  that  the  results  with  this  instrument 
require  corrections  for  systematic  errors  of  various  kinds.  But 
another  fortunate  result  is  that  the  determination  of  these 
corrections  is  always  possible,  and  when  observations  are  made 
on  a  system  so  nearly  uniform  through  long  periods  of  time,  the 
task  of  determining  and  applying  such  corrections  is  much  easier 


,§  181.]  THE  GREENWICH  OBSERVATIONS  343 

than  when  the  plan  of  work  is  frequently  changed.  With  all  its 
shortcomings,  the  Airy  transit  circle  has  proved  to  be  the  most 
serviceable  meridian  instrument  ever  constructed.  The  result 
is  that  the  Greenwich  observations  during  the  past  half  century 
afford  the  broadest  basis  we  now  possess  for  the  determination  of 
those  stars  of  which  accurate  positions  are  most  required. 

181.  The  German  school. 

Contemporaneous  with  the  accession  of  Pond  to  the  Director- 
ship of  the  Greenwich  Observatory  was  the  foundation  by 
Friedrich  Wilhelm  Bessel  of  the  German  school  of  practical 
astronomy.  The  fundamental  idea  of  this  school  in  the  trial  of 
its  instrument  reverses  the  maxim  of  English  criminal  law. 
The  instrument  is  indicted  as  it  were  for  every  possible  fault, 
and  is  not  exonerated  till  it  has  proved  itself  correct  in  every 
point.  The  methods  of  determining  the  possible  errors  of  an 
instrument  were  developed  by  Bessel  with  an  ingenuity  and 
precision  of  geometric  method  never  before  applied  to  such 
problems.  Not  only  this,  but  even  when  every  source  of  error 
admitting  of  determination  and  correction  has  been  allowed  for, 
the  instrumental  arrangement  must  admit  of  being  varied  from 
time  to  time  in  order  that,  if  any  undiscovered  errors  still  exist, 
they  may  be  detected  by  the  discrepancies  between  different 
methods  of  observation. 

Bessel's  fundamental  instrument,  after  a  few  years'  work 
with  an  old  circle  and  transit,  was  a  meridian  circle  con- 
structed by  Reichenbach.  Although  this  instrument  approached 
the  modern  form,  its  construction  was  far  from  perfect,  and, 
so  far  as  precision  of  individual  results  is  concerned,  it 
is  not  likely  that  the  work  of  Bessel  with  it  was  really 
superior  to  that  of  Pond.  The  far  greater  place  which  it  has 
filled  in  the  progress  of  practical  astronomy  is  due  more  to  the 
•excellence  of  Bessel's  system  of  reduction  and  discussion  than  to 
the  precision  of  the  instruments  and  observations  themselves. 
He  also,  as  compared  with  the  Astronomers  Royal  at  Greenwich, 
laboured  under  the  disadvantage  of  commanding  only  the  slender 
means  of  an  ill-endowed  observatory,  itself  only  an  adjunct  to  a 


344  OBSERVATIONS  OF  THE   STARS  [§182. 

university,  as   compared  with   the   resources   of  an   institution 
conducted  under  the  auspices  of  a  great  government. 

182.  The  Poulkova  Observatory. 

Bessel's  ablest  contemporary,  imbued  with  a  like  spirit,  and 
working  much  on  the  same  system,  was  Friedrich  Wilhelm 
Struve,  then  Director  of  the  Observatory  of  Dorpat.  It  became 
his  good  fortune  to  secure  the  support  of  a  powerful  government 
in  putting  into  practice  the  methods  of  the  German  school.  About 
1835  he  secured  from  the  Emperor  Nicolas  of  Russia  full 
authority  to  erect  and  equip  an  astronomical  observatory  of  the 
first  class,  which  should  be  a  credit  to  the  empire  to  which  it  be- 
longed. The.  new  establishment  was  erected  on  a  slight  eminence 
near  the  village  of  Poulkova,  18  kilometres  south  of  the  gate  of 
St.  Petersburg.  Struve's  special  purpose  was  to  introduce  a  new 
era  into  astronomical  determinations  by  combining,  on  a  large 
scale,  the  qualities  of  the  most  refined  instruments  that  art  could 
make  with  the  skill  of  the  most  capable  observers.  It  was  found 
that  when  the  latter  intelligently  devoted  the  most  painstaking 
attention  to  the  avoidance  of  every  source  of  error,  a  degree  of 
excellence  in  their  work  was  reached  which  was  not  possible 
with  mere  routine  observers. 

In  one  important  point  he  replaced  the  system  of  Bessel  by  an 
older  one.  For  the  determination  of  the  positions  of  the  funda- 
mental stars,  which  was  one  of  his  first  objects,  he  did  not  depend 
upon  the  meridian  circle,  by  which  observations  in  both  co- 
ordinates are  made  simultaneously,  but  constructed  the  transit 
instrument  and  vertical  circle  as  two  separate  instruments. 
The  vertical  circle  was  especially  designed  to  measure  zenith  dis- 
tances, not  only  at  the  moment  of  passing  the  meridian,  but  within 
a  short  distance  of  it  on  each  side.  In  the  hands  of  C.  A.  F.  Peters 
one  observation  with  this  instrument  was  worth  as  much  as 
twenty,  thirty,  or  even  forty  made  by  routine  observers  with  the 
meridian  circle. 

The  results  of  the  fundamental  determinations  of  star  places 
at  Poulkova  have  been  published  at  intervals  of  twenty  years, 
the  epochs  of  the  catalogues  having  been  1845,  1865,  and  1885. 


§  183.]  THE   POULKOVA  OBSEKVATORY.  345 

Latterly  the  number  of  stars  to  which  attention  is  devoted  at 
Poulkova  has  been  largely  increased,  one  of  the  works  being  a 
redetermination  of  the  positions  of  the  stars  observed  by  Bradley, 
more  than  three  thousand  in  number. 

The  two  great  observatories  of  Greenwich  and  Poulkova, 
through  their  rich  resources,  the  excellence  of  their  instruments, 
and  the  permanence  of  their  policy,  have  taken  the  leading  place 
in  supplying  material  for  the  fundamental  data  of  astronomy. 
The  other  national  observatories,  as  well  as  several  university 
observatories,  have  however  made,  from  time  to  time,  valuable 
contributions  toward  the  same  end.  Among  these  will  be  found 
the  national  observatories  at  Berlin,  Paris,  and  Washington,  and 
the  observatories  of  the  universities  of  Strassburg,  Abo,  Dorpat, 
Cambridge,  Edinburgh,  Glasgow,  and  Oxford,  as  well  as  those  of 
the  leading  American  Universities. 

There  are  other  observatories  whose  energies  have  been  directed 
rather  to  the  numerous  telescopic  stars  than  to  the  brighter 
fundamental  stars.  Here  the  Paris  Observatory  is  worthy  of 
especial  mention,  as  well  as  several  university  establishments, 
some  of  whose  publications  are  cited  in  the  notes  at  the  end  of 
the  present  chapter. 

183.  Observatories  of  the  southern  hemisphere. 

We  have  thus  far  glanced  only  at  the  observatories  of  the 
northern  hemisphere.  One  result  of  this  hemisphere  having  been 
the  first  seat  of  civilization  is  that  the  material  available  for  star 
determinations  in  the  southern  celestial  hemisphere  is  much  more 
scanty  than  for  the  northern.  The  observatories  of  Europe, 
being  generally  between  45°  and  60°  of  latitude,  have  not  been 
able  to  advantageously  extend  their  observations  to  more  than 
30°  of  south  declination.  Poulkova,  in  latitude  CO0,  is  most 
unfavourably  situated  in  this  respect.  The  parallel  of  30°  south 
is  on  its  horizon,  so  that  the  sun  at  the  winter  solstice  culmin- 
ates at  an  altitude  of  less  than  7°.  The  general  rule  is  that,, 
near  the  horizon,  vapours  in  the  atmosphere  detract  from  the 
accuracy  of  observations.  The  probable  error  of  any  astro- 
nomical observation  continually  increases  from  the  region  around 


346  OBSERVATIONS   OF  THE   STARS  [§  183. 

the  zenith  toward  the  horizon, — the  increase  being  very  slow 
down  to  an  altitude  of,  say,  30°, — but  being  more  and  more 
rapid  from  that  point  to  the  horizon  itself.  The  general  rule  is 
that  very  little  weight  can  be  assigned  to  observations  of  position 
&t  altitudes  less  than  10°.  The  Poulkova  observations  may  be 
regarded  as  somewhat  exceptional  in  this  respect,  since  the 
unusual  clearness  of  its  atmosphere  permits  observations  to  be 
advantageously  extended  nearer  to  the  horizon  than  is  possible 
in  less  favoured  regions.* 

Before  the  foundation  of  the  Cape  Observatory  the  observa- 
tions on  the  southern  stars  were  made  almost  entirely  by  indi- 
viduals at  establishments  more  or  less  temporary.  The  first 
•enterprise  of  this  kind  was  that  of  Lacaille,  who  was  sent  by  the 
French  authorities  to  the  Cape  of  Good  Hope  on  an  expedition 
for  astronomical  purposes  in  1750.  During  the  two  following 
years  he  made  observations  on  the  positions  of  nearly  ten 
thousand  stars  with  a  very  primitive  telescope,  used  to  make 
observations  on  zones  of  stars  with  the  instrument  in  a  fixed 
position  for  each  zone.  As  Lacaille  had  no  declination  micro- 
meter, it  was  not  possible  to  determine  zenith  distances  in  the 
usual  way  by  a  horizontal  thread,  and  the  ingenious  device 
of  a  rhomboid  was  adopted.  This  consisted  of  four  strips  of 
metal  placed  diagonally  in  the  field,  so  that  the  times  of  entrance 
-of  the  stars  into  the  rhomboid  and  of  their  exit  from  it  could  both 
be  observed  and  recorded.  The  mean  of  the  two  times  was  that 
of  the  transit  across,  the  middle  vertical  line  forming  one  diagonal 
of  the  rhomboid,  while  the  difference  of  the  times  showed  the 
distance  of  a  star  above  or  below  the  horizontal  diagonal. 

Naturally,  the  degree  of  precision  reached  by  this  method  was 
quite  low,  but  the  work  has  served  a  very  useful  purpose  during 
the  century  and  a  half  which  has  elapsed  since  its  completion. 
About  the  middle  of  the  last  century  a  reduction  of  Lacaille's 
observations  was  made  by  Baily  under  the  auspices  of  the 


*  The  author  was  once  informed  by  Otto  Struve,  then  Director  at  Poulkova, 
that  during  the  Crimean  War  he  could  see  through  the  great  telescope  the  men 
on  the  decks  of  the  British  fleet,  lying  off  Kronstadt,  at  a  distance  of  some 
25  miles. 


§  184.]  OBSERVATORIES  OF  THE  SOUTHERN  HEMISPHERE    347 

British  Association  for  the  Advancement  of  Science,  which  pub- 
lished the  results  in  a  catalogue. 

The  next  attack  on  the  southern  hemisphere  in  the  same 
direction  was  made  in  1822-26,  under  the  auspices  of  General 
Brisbane,  Governor  of  New  South  Wales,  by  Rumker.  Some 
years  later  the  observations  were  reduced  and  a  catalogue  of 
the  results  published  by  William  Richardson,  whose  name  the 
work  commonly  bears.  This  catalogue  includes  7835  stars. 

Shortly  afterward  Lieut.  Johnson,  at  St.  Helena,  made  a  com- 
paratively good  series  of  determinations  on  600  of  the  brighter  stars. 

The  observatory  of  the  Cape  of  Good  Hope  was  established  in 
1830.  Since  that  time  it  has  been  the  chief  centre  of  activity  in 
the  direction  now  under  consideration.  Its  instrumental  means 
and  the  methods  of  applying  them  have  continually  been 
improved,  until,  under  the  direction  of  Sir  David  Gill,  who  took 
charge  of  it  in  1880,  its  work  stands  second  in  excellence  to  that 
of  no  observatory  in  the  world. 

184.  Miscellaneous  observations. 

The  work  at  Greenwich  and  Poulkova  is,  so  far  as  positions  of 
the  fundamental  stars  are  concerned,  pre-eminent  for  its  syste- 
matic character  and  for  the  long  period  through  which  it  has 
extended.  But  many  important  though  more  temporary  works 
have  been  carried  out  for  the  same  purpose.  First  in  the  order 
of  time  must  be  mentioned  that  of  Piazzi  at  Palermo,  who,  before 
and  after  the  beginning  of  the  nineteenth  century,  made  a  long 
series  of  observations  on  the  fundamental  stars,  employing  the 
transit  instrument  and  the  complete  circle.  His  declinations  of 
the  fundamental  stars  were  almost  or  quite  the  first  made  with 
the  latter  instrument.  The  observations  were  not,  however, 
confined  to  the  fundamental  stars,  but  included  nearly  ten 
thousand  stars  of  all  the  brighter  magnitudes.  Unfortunately 
only  the  imperfect  reductions  of  Piazzi  himself  have  as  yet  been 
available  for  the  use  of  the  astronomer;  but  a  reduction  with 
modern  data  and  by  modern  methods  is  now  being  executed 
under  the  auspices  of  the  Carnegie  Institution  by  Dr.  Herman  S. 
Davis. 


348  OBSERVATIONS   OF  THE   STARS  [§184. 

The  works  of  the  same  class  by  Struve  at  Dorpat,  before  his 
removal  to  Poulkova,  Argelander  at  Abo,  Airy  and  his  successors 
at  Cambridge,  various  observers  at  Washington,  as  well  as 
numerous  lesser  works  of  various  observers  will  be  cited  here- 
after. 

185.  Observations  of  miscellaneous  stars. 

The  observations  thus  far  reviewed  are  mainly  those  leading 
to  determinations  of  the  positions  of  fundamental  stars.  As 
already  remarked,  the  distinction  between  stars  which  are 
fundamental  and  those  which  are  not  is  somewhat  vague.  Yet, 
a  fairly  definite  line  may  be  drawn  between  observations  leading 
to  independent  and  accurate  determinations  of  a  limited  number 
of  stars,  and  those  in  which  precision  is  sacrificed  in  order  to 
extend  the  work  to  a  larger  number  of  minute  stars.  Obser- 
vations of  the  former  class  have  been  mainly  of  the  kind  which 
in  the  preceding  chapter  were  defined  as  fundamental,  those  of  the 
latter  class  as  differential  It  may  also  be  remarked  that  the 
determinations  of  the  first  class  have  been  mostly  confined  to  a 
few  thousand  of  the  brighter  stars,  including  the  most  of  those 
to  the  sixth  magnitude,  and  occasionally  a  few  of  the  seventh  or 
eighth. 

During  the  time  of  Bradley,  and  of  his  immediate  successors,, 
no  attempt  was  made  to  determine  the  positions  of  the  innumer- 
able faint  telescopic  stars  which  stud  the  heavens,  nor  even  to 
catalogue  them.  The  first  serious  work  in  this  direction  was 
that  of  Lalande  at  Paris,  who,  near  the  close  of  the  eighteenth 
century,  and  therefore  contemporaneously  with  Piazzi,  undertook 
the  work  of  cataloguing  all  the  stars  visible  in  his  instrument 
with  a  completeness  which  has  scarcely  been  exceeded  even  up 
to  the  present  day.  His  Histoire  Cdleste,  which  is  in  use  even 
now  in  determining  the  proper  motions  of  the  fainter  stars,  is  a 
long  enduring  monument  to  his  industry.  His  energies  were  so 
entirely  absorbed  in  the  work  of  observation  that  he  made  no 
serious  attempt  at  their  complete  systematic  reduction.  This 
much  needed  work  was  carried  out  by  Francis  Baily  of  England 
in  1840,  and  published  in  a  thick  volume  containing  the  reduced 


§  185.]       OBSERVATIONS   OF  MISCELLANEOUS   STABS  349 

positions  of  the  stars  observed  by  Lalande.  But  the  data  and 
methods  of  reduction  then  available  were  both  imperfect,  and 
the  need  of  a  complete  rereduction  has  not  yet  been  supplied. 
The  best  approach  to  it  is  found  in  a  set  of  tables  by  von  As  ten, 
by  which  the  astronomer  may  rapidly  reduce  any  of  Lalande's 
observations  with  data,  which,  although  better  than  those  used 
by  Baily,  do  not  fully  satisfy  modern  requirements. 

When  the  object  is  to  determine  the  positions  of  the  greatest 
possible  number  of  stars,  the  observations  have  to  be  made  by 
zones  of  declination,  of  greater  or  less  breadth,  according  to  the 
requirements  of  the  case.  The  advantage  of  the  zone  system  is 
that  the  observer  does  not  have  to  move  his  telescope  through 
wide  arcs  of  declination  between  one  observation  and  the  next, 
but  keeps  it  in  nearly  the  same  position  during  a  period  of 
several  hours,  perhaps  through  a  whole  night's  work.  The 
breadth  of  the  zone  depends  upon  the  number  of  stars  which 
it  is  desired  to  include.  Sometimes  the  system  has  been  to  keep 
the  telescope  in  a  fixed  position  through  the  entire  course  of  an 
evening,  observing  as  many  as  possible  of  the  stars  which  pass 
over  the  threads  in  the  focus.  As  the  field  of  view  cannot 
advantageously  take  in  more  than  a  fraction  of  a  degree,  this 
system  is  a  very  slow  one  if  it  is  intended  to  cover  the  entire 
sky.  The  more  usual  proceeding  has,  therefore,  been  to  include 
a  zone  a  few  degrees  wide,  generally  less  than  5°,  in  each  night's 
work. 

Lalande's  observations  were  made  on  this  system.  Twenty 
years  later  zone  observations  covering  a  considerable  portion 
of  the  northern  sky  were  commenced  by  Bessel  at  Konigsberg 
and  Argelander  at  Abo.  The  results  of  these  two  works  have 
been  collected  by  Weisse  and  Oeltzen  in  well-known  catalogues. 

In  1846  a  series  of  zone  observations,  intended  to  ultimately 
cover  as  much  as  possible  of  the  sky,  was  commenced  at  the 
Naval  Observatory  at  Washington.  But  the  work  was  left 
unfinished  after  two  or  three  years.  The  observations  have 
been  published,  but  not  in  complete  form.  Some  of  them 
labour  under  the  disadvantage  of  having  been  made  by  inex- 
perienced observers  who  made  many  errors  in  writing  down 


350  OBSEEVATIONS  OF  THE   STARS  [§  185. 

the  numbers  of  their  records,  while  others  are  of  the  best 
class. 

About  1850  Bond,  at  the  Harvard  Observatory,  observed  a 
few  zones  near  the  equator,  which  are  quite  unique  in  astronomy. 
They  were  made  with  the  great  equatorial,  which  was  clamped 
in  a  fixed  position  for  each  strip  observed,  and  the  R.A.'s  and 
Decs,  of  the  stars  determined  as  they  passed  through  the  field. 
The  results,  which  are  found  in  the  early  volumes  of  the  Harvard 
Annals,  extend  to  fainter  stars  than  have  since  been  catalogued 
or  even  listed.  They  will,  however,  be  included  in  the  Inter- 
national Chart  of  the  heavens  now  being  made  by  photography 
at  a  large  number  of  observatories  in  both  hemispheres.  Prob- 
ably all  those  actually  observed  by  Bond  will  be  in  the 
equatorial  zone,  which  is  being  photographed  at  the  Observatory 
of  Algiers. 

When  Le  Verrier  took  charge  of  the  Paris  Observatory  in 
1853,  one  of  the  projects  which  he  instituted  was  that  of  the 
redetermination  of  all  of  Lalande's  stars.  This  work  was 
completed,  and  the  results  have  been  published  during  the 
past  ten  years  in  a  catalogue  filling  four  large  quarto  volumes. 
This  immense  work  includes,  not  only  the  fainter  stars  observed 
by  Lalande,  but  nearly  all  the  stars  of  the  brighter  classes.  But 
the  method  of  reduction  adopted  by  Le  Verrier,  and  pursued 
since  his  time  at  Paris,  is  far  from  the  best.  Not  only  is  all 
the  work  purely  differential,  both  in  R.A.  and  Dec.,  but  sufficient 
attention  has  not  been  paid  to  the  sources  of  systematic  error  to 
which  such  work  is  liable,  especially  in  the  adopted  positions 
of  the  standard  stars.  The  result  is  that,  at  the  best,  it  is 
scarcely  possible  to  apply  any  systematic  corrections  which 
will  not  leave  accidental  errors  outstanding  larger  than  should 
be  found  in  results  of  the  highest  class.  This  does  not  greatly 
diminish  the  value  of  the  work  for  the  faint  Lalande  stars,  but 
does  for  the  brighter  stars. 

In  1865  the  Astronomische  Gesellschaft,  an  international 
association,  having  its  headquarters  in  Germany,  formed  the 
design  of  a  complete  redetermination  of  the  positions  of  all 
the  stars  in  the  northern  celestial  hemisphere,  those  within 


§  186.]       OBSERVATIONS   OF  MISCELLANEOUS   STARS  351 

10°  of  the  pole  excepted,  down  to  the  ninth  magnitude,  with 
as  near  an  approach  as  possible  to  the  modern  standard  of 
precision.  With  a  single  gap,  the  results  of  this  work  have 
all  been  published  in  the  several  catalogues  issued  by  the 
society. 

When  the  observations  for  this  work  were  approaching  com- 
pletion, the  project  of  extending  it  to  23°  south  declination 
was  undertaken,  and  catalogues  down  to  this  point  are  now  in 
process  of  preparation  and  publication. 

Durchmusterungen.  The  star-lists,  familiarly  known  as- 
"  Durchmusterungen,"  belong  to  a  different  class  from  the 
preceding,  not  being  intended  to  record  the  accurate  position 
of  a  star,  but  only  its  approximate  position  with  sufficient 
precision  to  enable  the  star  to  be  certainly  identified.  The  first 
comprehensive  work  of  this  sort  was  carried  on  at  Bonn  by 
Argelander  and  Schonfeld,  and  extended  from  the  North  Pole 
to  —1°  of  south  declination. 

Schonfeld  afterwards  extended  the  work  to  —23°.  From  this- 
parallel  to  the  south  pole  a  Durchmusterung  is  being  carried  on 
in  the  Argentine  Republic,  at  the  Cordova  Observatory,  by  Thome, 
who  has  published  three  volumes  of  it,  extending  to  —53°  of 
declination. 

The  Cape  Photographic  Durchmusterung  by  Gill  and  Kapteyn, 
is  based  on  photographs  of  the  sky  taken  at  the  observatory  of 
the  Cape  of  Good  Hope,  and  covers  the  sky  from  —18°  to  the 
South  Pole.  This  is  the  best  arranged  and  digested  work  of  its- 
class  that  has  yet  appeared.  The  number  of  stars  included  in 
it  is  not  however  so  great  as  in  the  Cordova  work,  and  the 
magnitudes  assigned  to  the  stars  are  subject  to  revision. 

Section  II.  Reduction  of  Catalogue  Positions  of  Stars  to  a 
Homogeneous  System. 

186.  Systematic  differences  between  catalogues. 

The  leading  observatories  at  which  meridian  observations  are 
made  collect  the  results  from  time  to  time  in  the  form  of 
catalogues  of  the  positions  of  stars.  Generally  such  a  catalogue 


352  CATALOGUES   OF  THE   STABS  [§  186. 

is  given  in  connection  with  the  observations  of  each  year.  After 
-a  number  of  years  the  annual  results  thus  derived  may  be  com- 
bined in  a  single  catalogue,  in  which  the  mean  positions  of  the 
stars  are  reduced  to  some  common  equinox  of  reference.  Posi- 
tions of  the  brighter  and  more  important  stars  are  generally 
contained  in  quite  a  number  of  these  catalogues,  and  when  the 
definitive  position  of  a  star  is  to  be  worked  out,  the  best  result  is 
reached  by  combining  the  data  of  all  the  catalogues  in  which 
it  occurs. 

When  we  compare  the  mean  position  of  a  star  for  any  one 
epoch,  as  found  in  different  catalogues,  we  are  to  expect  dif- 
ferences arising  from  the  accidental  and  unavoidable  errors  of 
observation,  which  it  is  desirable  to  eliminate  by  combining  as 
many  authorities  as  possible.  If  we  represent  by  Sv  $2,  J3 ...  the 
differences  between  the  coordinates  of  the  same  stars  in  one 
catalogue  A,  and  in  another  catalogue  B,  the  mean  of  these 
differences  should,  if  they  were  wholly  in  the  nature  of  indepen- 
dent and  fortuitous  errors,  converge  toward  zero  as  their  number 
increases.  But,  as  a  matter  of  fact,  this  is  seldom  found  to  be 
the  case.  To  fix  the  ideas,  let  us  suppose  a  comparison  to  be 
made  between  all  the  declinations  in  a  zone  5°  wide.  We  shall 
nearly  always  find  that  the  mean  value  of  the  differences,  as 
the  number  of  comparisons  is  increased,  converges  toward  some 
well-marked  positive  or  negative  value,  and  not  towards  zero,  as 
it  should  if  the  errors  of  each  authority  were  purely  accidental, 
in  comparing  even  the  best  catalogues  this  value  may  be  several 
tenths  of  a  second.  This  shows  that,  in  addition  to  the  accidental 
errors  necessarily  affecting  all  astronomical  determinations,  there 
must  be  some  source  of  systematic  error  affecting  the  positions 
in  one  catalogue  differently  from  those  in  another.  The  possible 
sources  of  such  errors  are  many.  They  may  be  classified  as 
follows : 

1.  Differences  in  the  methods  and  data  of  reduction.  Examples 
of  this  class  are  differences  in  the  adopted  value  of  the  constants 
of  nutation  and  aberration ;  differences  in  the  adopted  positions 
of  the  fundamental  stars  used  for  clock  errors ;  differences  in  the 
•constant  of  refraction;  the  employment  of  an  erroneous  latitude 


$  186.]  DIFFERENCES   BETWEEN  CATALOGUES  353 

or  non-correspondence  of  the  adopted  latitude  with  that  given 
by  the  nadir  point  of  the  instrument.  All  differences  of  this 
class  admit  of  being  reconciled  by  applying  to  the  positions 
given  in  the  catalogues  the  corrections  necessary  to  reduce  the 
results  to  what  they  would  have  been  had  the  same  data  and 
methods  of  reduction  been  used  in  each.  The  extent  to  which 
it  is  advisable  to  apply  these  corrections  must  depend  on  the 
labour  involved  and  value  of  the  results  to  be  reached. 

2.  Causes  of  systematic   error   which   we    know   may  have 
affected   the    observations,   but   of   which   we    have   no  way   of 
estimating  the  magnitude.     One  example  of  this  arises  where 
the  errors  of  graduation  of  a  circle  have  not  been  investigated  at 
all,  or  have  not  been  well  determined.     In  such  cases  all  the 
declinations  may  be  in  error  by  amounts  varying  with  the  zenith 
distance ;  but  it  will  be  impossible   to  determine  the  amount 
except  by  comparison  with  other  authorities.     Another  example 
is  the  possible  difference  between  day  and  night  observations, 
arising  from  the  different  conditions  under  which  they  are  made. 
We  may  take  it  for  granted  that  the  diurnal  changes  of  tempera- 
ture of  an  instrument  and  its  surroundings  may  vary  the  adjust- 
ments in  a  way  not  admitting  of  determination,  yet  following  a 
general   law  which  may   be  expressed  in  a  formula   of   which 
the  numerical   elements  can  be  inferred  from  a  comparison  of 
authorities.     The  personal  equation  of  an  observer  may  also  be 
different   for   stars   observed    by  day   and    by    night.      In   this 
case  it  will  be  possible  to  formulate  some  sort  of  a  law  which 
the  errors  should   follow ;  but  the   determination   of  the  exact 
amount  of  the  error  will  not  be  possible  except  by  a  comparison 
of  results. 

3.  Errors  arising  from  unknown   causes   which  elude  both 
investigation  and  exact  statement.     We  find  by  comparing  the 
independent  results  of  the  work  of  different  observatories  that 
differences    show    themselves    which    we    may    attribute    to    a 
number  of  causes,  temporary  or  variable,  the  results  of  which  do 
not  admit  of  being  reduced  to  any  well-defined  law.     The  causes 
of  the  errors  thus  indicated  may  be  temporary, — varying  from 
one    night   to    another — or   they   may   affect    the   work   of   an 

N.S.A.  z 


354  CATALOGUES  OF  THE   STARS  [§  186. 

observer  or  an  instrument  through  months  or  years.  They  can 
be  determined  in  each  case  only  by  an  intercomparison  of  inde- 
pendent authorities. 

Until  recent  times  little  or  no  account  was  taken  of  these 
systematic  differences  among  catalogues  of  stars.  When  the 
position  and  proper  motion  of  a  star  were  to  be  determined  from 
a  combination  of  observations  or  of  catalogue  positions,  the 
deviations  of  the  several  catalogues  from  the  general  mean  were 
regarded  as  purely  accidental  errors.  The  practice  of  correcting 
catalogues  for  systematic  differences  among  them  is  largely  due 
to  the  example  and  researches  of  Auwers,  which  date  from  1865. 
In  explaining  the  system  of  correction  we  must  premise  that  the 
processes  are  largely  tentative,  and  that,  from  the  very  nature  of 
the  case,  the  exact  values  of  the  corrections  must  remain  more  or 
less  in  doubt. 

187.  Systematic  corrections  to  catalogue  positions. 

The  general  idea  on  which  systematic  corrections  of  the  kind 
in  question  are  to  be  derived  is  that,  in  each  individual  catalogue, 
whole  groups  of  stars  may  be  affected  by  common  sources  of 
error  peculiar  to  the  group.  For  example,  there  are  many  causes 
which  may  affect  all  stars  of  the  same  declination  with  one  and 
the  same  constant  error.  Other  causes  may  affect  all  the  stars  of 
one  Right  Ascension  in  the  same  way,  but  stars  of  different 
Right  Ascensions  in  different  ways.  Apart  from  errors  in  the 
elements  of  reduction,  refraction,  nutation,  aberration,  etc.,  these 
errors  can  be  brought  out  only  by  a  comparison  of  separate  and 
independent  authorities  inter  se.  The  idea  is  then  to  apply  to 
positions  given  by  each  separate  authority  such  corrections  for 
the  several  groups  of  stars  as  shall  bring  the  whole  into  general 
harmony.  In  doing  this  different  weights  will  be  assigned  to 
different  authorities  according  to  the  supposed  freedom  of  the 
results  from  sources  of  systematic  error.  When  the  positions  of 
each  authority  are  thus  corrected  the  discrepancies  still  outstand- 
ing should  follow  the  law  of  purely  fortuitous  errors. 

It  should  be  remarked  that  the  relative  weights  assigned  to 
the  different  authorities  for  the  purpose  of  this  combination  may 


§  187.]          CORRECTIONS  TO  CATALOGUE   POSITIONS  355 

be  very  different  from  those  to  which  the  positions  of  individual 
stars  are  entitled  after  the  corrections  have  been  made.  For 
example,  so  far  as  freedom  from  systematic  errors  of  Class  B  is 
concerned,  the  positions  of  stars  obtained  at  Greenwich  and  at 
Poulkova  may  be  entitled  to  equal  weight.  But  the  probable 
deviation  of  one  observation  of  declination  at  Greenwich  is  more 
than  double  that  of  one  observed  at  Poulkova.  In  consequence 
the  weight  to  be  assigned  to  a  single  observation  at  Poulkova  is 
more  than  four  times  that  assigned  at  Greenwich. 

When  a  S3rstem  of  corrections  is  constructed  for  a  number  of 
standard  catalogues,  the  harmonious  set  of  positions  to  which 
all  the  catalogues  are  reduced  is  known  as  a  fundamental 
system.  Such  a  system  may  be  considered  as  embodied  in  a 
more  or  less  complete  set  of  corrections  of  the  kind  described,  or 
in  a  standard  catalogue  of  stars  constructed  from  all  available 
observations  after  systematic  corrections  have  been  applied  to 
the  individual  authorities. 

The  purpose  being  to  find,  not  corrections  to  individual  stars, 
but  the  mean  corrections  to  whole  groups  of  stars,  two  methods 
may  be  adopted.  One  method  consists  in  comparing  the  positions 
of  standard  stars  in  each  separate  catalogue  with  some  one  stand- 
ard catalogue.  It  is  not  necessary,  in  the  first  place,  that  this 
standard  of  comparison  should  be  a  definitive  one,  because  the 
correction  of  the  standard  itself  will  always  be  in  view.  Let  us 
suppose,  to  fix  the  ideas,  that  the  adopted  standard  positions  of 
the  fundamental  stars  in  a  certain  zone  of  declination,  5°  or  10° 
in  breadth,  are  compared  with  those  found  for  the  same  epoch 
from  all  the  other  available  catalogues.  Let  the  mean  deviations 
of  the  standard  thus  found  be : 

From  the  standard  itself  zero 
„      Catalogue  A  Sl 


Then,  by  applying  the  several  corrections  Slt  32,  etc.,  to  the 
coordinates  in  the  several  catalogues,  they  will  all  be  brought 
into  harmony  with  the  standard. 


356  CATALOGUES  OF  THE   STARS  [§  187. 

But  the  standard  itself  may  need  correction,  because  the 
proper  standard  is  the  general  mean  of  all.  To  find  this  mean 
let  the  weights  assigned  to  the  several  catalogues  be 


W0  being  the  weight  of  the  provisional  standard  itself.  Then,  the 
weighted  mean 

A     = 

will  be  the  deviation  of  the  standard  from  the  mean  of  all  the 
authorities,  itself  included.  Since  this  last  mean  is  the  final 
standard,  it  follows  that  A  will  be  the  deviation  of  S  from 
this  final  standard,  and  —A  the  correction  to  reduce  the  standard 
to  the  general  mean.  The  corrections  then  necessary  to  reduce 
the  several  catalogues  to  the  mean  standard  are  : 

Standard  catalogue  ;       corr.  =  —  A. 
A  i      „     =^-A. 


the  weighted  mean  of  which  is  zero,  as  it  should  be. 

The  preceding  method,  if  applied  without  modification,  is 
subject  to  the  drawback  that  it  is  not  easy  to  find  any  one 
catalogue  sufficiently  complete  and  comprehensive  to  serve  as 
the  sole  basis  of  comparison  with  all  other  catalogues  at 
all  epochs.  The  method  may,  therefore,  be  modified,  as  necessary, 
by  comparing  pairs  of  catalogues  A  and  B  for  the  same  epoch. 
If  Sl  and  (?2  be  the  unknown  corrections  of  the  two  catalogues 
for  any  zone  or  region,  and  A1>2  their  mean  difference  within 
this  zone  or  region  we  shall  have 

*!-*,  =  A,.,. 

We  may  then  reduce  all  the  comparisons  of  either  of  the  two 
catalogues  with  the  standard  =S  to  the  mean  of  the  other,  and 
thus  gain  for  each  the  benefit  of  the  comparisons  of  S  with  the 
other.  For  example,  assume  the  case  that  we  have  6  stars 
common  to  A  and  S,  and  4  common  to  B  and  S ;  but  a  much 


§  188.]         CORRECTIONS  TO  CATALOGUE  POSITIONS  357 

greater  Dumber  common  to  A  and  B.     From  the  comparison  of 
A  and  B  we  find  a  value  of  Ali2  =  4  —  B.     Let  us  also  put 

ev  the  mean  of  the  6  differences  S  —  A  ; 
<?2,  the  mean  of  the  4  differences  S  —  B. 

We  shall  then  have,  for  Sl  =  S  —  A,  two  independent  values, 
namely 

1.  That  from  direct  comparison  :  Sl  =  ev 

2.  That  through  B:  ^i  =  e2  +  Ai,2- 

The   concluded   values   of  ^  and  <52  are   then    the    weighted 
mean  of  these  two  values,  giving 


l~  10 

In  a  similar  way  we  have  for  &,  the  two  values  e2  and  e1  —  A1>2, 

,  _6(e1-A,2)+4e2 
~10~ 

188.  Form  of  the  systematic  corrections. 

We  have  now  to  consider  the  general  form  of  the  systematic 
corrections  ordinarily  applied.  It  is  common  to  regard  them 
as  two  in  number  for  each  coordinate,  one  being  a  function 
of  the  Right  Ascension  alone;  the  other  of  the  Declination 
alone.  The  designations  of  the  corrections  are  as  follows  : 

Aota,  correction  to  the  R.A.  depending  on  the  R.A.  ; 
Aocs,  correction  to  the  R.A.  depending  on  the  Declination  ; 
A<5tt,  correction  to  the  Declination  depending  on  the  R.A.  ; 
A(55,  correction  to  the  Declination  depending  on  the  Dec. 

The  necessity  for  the  correction  Aaa  has  arisen  through  the 
wide  adoption  of  an  erroneous  system  of  Right  Ascensions  dating 
from  the  time  of  Pond,  which  were  transplanted  by  Le  Verrier 
in  his  fundamental  catalogue  used  in  reducing  the  Paris  obser- 
vations, and  used  by  Airy  in  the  early  Greenwich  work.  By 
successive  revisions  of  the  fundamental  Right  Ascensions  this 
error  may  be  gradually  reduced,  the  rapidity  of  the  reduction 
depending  on  the  number  of  hours  through  which  observations 


358  CATALOGUES   OF  THE   STARS  [§  188. 

extend  during  each  night.  What  remains  of  the  error  is  in 
this  way  smoothed  off  to  the  general  form 

Aoca  =  a  cos  R.A.  +  &  sin  RA. 

In  all  good  catalogues  it  should  be  assumed  as  of  this  form,  if  it 
exists  at  all.  As  a  matter  of  fact,  however,  it  has  been  nearly 
or  quite  smoothed  out  in  the  recent  Greenwich  catalogues,  and 
in  most  other  modern  catalogues,  except  those  based  on  the 
Paris  Right  Ascensions. 

It  may  be  remarked  that  Auwers  does  not  assume  the  cor- 
rection to  be  of  this  form,  but  determines  it  from  hour  to  hour, 
smoothing  off  the  results  so  that  they  shall  be  represented  by  a 
regular  curve. 

The  error  which  Aafi  is  intended  to  eliminate  has  arisen 
mainly  from  the  pivots  of  the  transit  instrument  not  being 
perfect  cylinders,  whereby  the  line  of  collimation  slightly 
deviates  to  one  side  or  the  other  of  the  meridian  of  the 
instrument.  An  erroneous  determination  of  the  error  of  col- 
limation will  also  produce  an  error  of  this  sort.  Near  the 
pole  other  causes,  due  to  personal  equation,  come  into  play. 
The  personal  equation  of  the  observer  is  very  likely  to  be 
different  for  a  slow  moving  star  than  for  the  rapidly  moving 
equatorial  stars.  Moreover,  close  circum polar  stars  can  be  better 
observed  by  eye  and  ear  than  by  the  chronograph.  But  when 
the  latter  is  used  for  the  quick  moving  stars,  while  those  near 
the  pole  are  observed  by  eye  and  ear,  there  is  likely  to  be  a 
change  in  the  personal  equation  from  one  cla*ss  to  the  other. 

The  Right  Ascensions  of  the  catalogue  may,  as  already  shown, 
also  require  a  constant  equinoctial  correction.  This  may  be 
combined  either  with  Aoca  or  Aafi.  The  former  is  the  more 
common  method,  since  the  correction  is  in  this  way  very  easy 
to  determine.  Auwers,  however,  adds  it  to  the  values  of  Ao.g, 
which  on  the  score  of  exactness  is  the  preferable  system. 

189.  Method  of  finding  the  corrections. 

The  practical  method  of  forming  the  tables  for  Aoca  and  AcSs  is 
to  begin  the  comparison  by  zones  of  declination.  A  zone  of 


§  190.]         METHOD  OF  FINDING  THE   CORRECTIONS  359 

declination  of  suitable  breadth,  generally  5°  or  10°,  is  taken, 
and  for  all  the  stars  within  it  common  to  the  two  catalogues 
the  differences  between  the  fundamental  catalogue  and  that  to 
be  corrected  are  formed  for  each  coordinate,  and  the  mean  taken. 
This  mean  will  be  the  preliminary  value  of  Aoc5  or  A6*«  for  the 
middle  declination  of  the  zone.  This  process  being  extended 
to  all  the  zones  in  the  catalogue,  we  shall  have  values  of  the 
correction  for  each  zone. 

These  corrections  are  then  to  be  arranged  in  a  column  and 
smoothed  off,  so  as  to  form  the  ordinates  of  a  regularly  varying 
curve,  and  interpolated  to  every  round  5°  or  10°  of  declination, 
as  the  case  may  be,  with  the  declination  as  an  argument. 

To  form  the  corrections  depending  on  the  R.A.,  the  catalogue 
places  should  first  be  corrected  for  the  terms  depending  on  the 
Dec.,  and  the  corrected  places  compared  with  those  of  the 
standard  catalogue.  The  mean  of  the  residual  differences  is  then 
taken  for  every  lh,  Zh,  or  3^,  of  R.A.,  smoothed  off,  and  arranged 
in  a  table  with  the  RA.  as  the  argument. 

In  the  formation  of  Aoca  it  is  advisable  to  give  most  weight  to 
stars  within  30°  of  the  equator,  and  little  or  no  weight  to  stars 
north  of  50°,  or  perhaps  60°.  The  exact  method  of  proceed- 
ing must  depend  on  the  peculiarities  of  the  catalogue  to 
be  reduced,  especially  on  the  methods  of  observation  and 
reduction. 

190.  Distinction  of  systematic  from  fortuitous  differences. 

An  important  question  in  preparing  such  tables  is  how  far  the 
differences  which  we  find  should  be  regarded  as  accidental  rather 
than  systematic  and  should,  therefore,  be  ignored  as  arising  from 
fortuitous  errors.  By  the  theory  of  these  errors,  when  the  mean 
of  a  number  of  them  is  taken,  the  probable  value  of  this  mean 
will  diminish  as  the  square  root  of  the  number  whose  mean  is 
taken.  We  may  therefore  say,  in  a  general  way,  that  if  the 
mean  systematic  correction  thus  derived  does  not  exceed  the 
probable  mean  of  the  fortuitous  deviations,  we  should  disregard 
it.  The  same  may  be  true  even  should  the  mean  value  be 
greater  than  that  set  by  the  limit.  That  is  to  say,  if  we  put 


360  CATALOGUES  OF  THE   STARS  [§  190. 

e,  the   general   mean  value  of  the  differences  whose  mean 

is  taken ; 
N,  the  number  of  these  differences ; 

Then,  if  the  final  mean  is  less  than  —/=,  there  is  no  reason  to 

+JN 

regard  the  differences  as  systematic.  And  even  should  it  exceed 
this  limit,  the  reality  of  the  difference  may  be  in  doubt.  In  any 
case  we  should  see  whether  the  differences  remain  of  the  same 
sign  through  several  zones  of  declination.  If  they  do,  the  reason 
for  considering  them  real  is  strengthened,  if  not  it  is  weakened. 
In  view  of  the  probable  amount  of  the  accidental  deviations,  the 
rule  should  be,  when  in  doubt  as  to  the  amount  of  a  correction, 
to  assign  it  the  smallest  probable  value. 

From  this  point  of  view  it  seems  quite  likely  that  many  of  the 
systematic  corrections  found  in  existing  tables  should  be  re- 
garded either  as  unreal  or  as  being  too  large.  It  may  also  well 
be  that  they  vary  too  rapidly  from  one  zone  of  Declination,  or 
hour  of  Right  Ascension  to  another. 

Practically,  it  will  seldom  be  necessary  to  construct  new  tables 
or  corrections  for  any  of  the  fundamental  catalogues,  because 
such  tables  have  already  been  constructed  by  Auwers,  Boss  and 
the  present  writer,  and  can  be  used  to  such  an  extent,  or  com- 
bined in  such  a  way  as  the  computer  deems  best.  Every  new 
catalogue  that  appears  will,  however,  need  examination  with  a 
view  of  constructing  tables  for  the  purpose  in  question,  and  it  is 
in  this  case  that  the  preceding  methods  and  principles  have  to  be 
applied. 

191.  Existing  fundamental  systems. 

A  "  fundamental  system  "  of  star-positions  may  be  defined  in 
either  of  two  ways.  One  is  by  a  sufficiently  extensive  catalogue 
of  fundamental  stars,  in  which  the  position  and  proper  motion 
of  each  individual  star  has  been  worked  out  with  the  greatest 
possible  precision.  It  is  then  assumed  that  the  errors  of  such 
a  catalogue,  both  accidental  and  systematical,  are  as  small  as 
possible,  and  that  the  latter  vary  very  slowly  from  one  region  of 
the  sky  to  another. 


191.]  EXISTING   FUNDAMENTAL   SYSTEMS  361 

A  fundamental  system  may  also  be  defined  by  tables  of  cor- 
rections of  the  form  just  explained  for  the  best  existing  star 
catalogues.  For  by  applying  these  corrections  to  catalogue 
positions  of  the  stars,  positions  and  proper  motions  of  each 
individual  star  are  obtained  which  will  be  in  harmony  with  the 
fundamental  system  on  which  the  corrections  are  based. 

The  following  are  the  fundamental  systems  which  have  been 
most  used. 

1.  The  system  constructed  by  Auwers  for  use  in  reducing  the 
observations  of  zones  of  stars  made  under  the  auspices  of  the 
Astronomische  Gesellschaft.  It  is,  therefore,  commonly  known 
as  the  "A.G.  System."  The  star  positions  which  form  it  are 
found  in  No.  xiv.  of  the  Publicationen  der  Astronomischen 
Gesellschaft,  and  an  extension  into  the  southern  hemisphere  in 
Publication  xvii.  of  the  same  series. 

In  this  catalogue  the  modern  positions,  generally  for  a  mean 
epoch  about  1865,  were  derived  from  a  careful  discussion  and 
combination  of  all  the  best  modern  determinations.  The  proper 
motions,  which  are  an  essential  part  of  any  fundamental  system^ 
were  found  by  a  comparison  of  the  observations  of  Bradley 
(mean  epoch  about  1755)  with  a  preliminary  (not  with  the 
definitive)  catalogue  of  modern  positions,  and  were  not  further 
altered. 

The  result  of  this  is  that,  if  the  fundamental  places  are  reduced 
back  to  1755,  they  will  not  rigorously  agree  with  Auwers- 
Bradley,  but  will  deviate  by  an  amount  equal  to  the  correction 
applied  to  reduce  the  modern  provisional  place  to  the  definitive 
place. 

It  is  now  well  established  that  the  Bradley  positions  are 
affected  by  considerable  systematic  errors.  The  consequence  of 
this  is  that,  through  the  proper  motions,  the  A.G.  system  is,  for 
our  epoch,  affected  by  systematic  errors  of  the  opposite  algebraic 
sign,  increasing  uniformly  with  the  time. 

2.  At  nearly  the  same  time  as  the  A.G.  system  appeared 
the  system  of  Professor  Lewis  Boss.  This  included  only  the 
declinations.  The  proper  motions  were  derived  by  the  rigorous 
process  of  a  least  square  solution  of  all  the  results  of  observations. 


362  CATALOGUES  OF  THE  STARS  [§  191. 

This  system  was  adopted  in  the  American  Ephemeris  from  1883 
to  1899,  and  was  employed  in  the  researches  made  in  the  office 
of  that  work  up  to  1897. 

The  declinations  were  so  thoroughly  worked  up  by  Boss  that 
the  continued  use  of  the  A.G.  system  of  declinations  until  1900 
is  to  be  regarded  as  unfortunate. 

3.  The  third  system  is  that  of  Newcomb,  found  in  the 
Astronomical  Papers  of  the  American  Ephemeris,  vol.  viii. 
In  forming  the  fundamental  declinations  the  processes  show  the 
following  main  features. 

(a)  The  systematic  corrections  to  the  Boss  system  were  found 
in  advance  of  determining  places  of  the  individual  stars. 

(b)  In   the   case   of  each  instrument  used    in   forming   these 
systematic  corrections,  the  error  of  its  declinations  of  stars  near 
the  equator  was  determined  by  the  general  principle  that  the 
planets  move  around  the  sun  in  great  circles.     In  consequence, 
the  declinations  of  the  planets  are,  in  the  general  mean,  to  be 
regarded  as  absolutely  correct.     Accordingly,  if  the  declinations 
of  any  planet,  through  an  entire  revolution,  are  found,  by  instru- 
mental measurement  or  by  comparison  with  the  declinations  of 
a  fundamental  catalogue,  to  be  in  error  by  the  constant  quantity 
€,  we  conclude  that  this  error  is  not  real,  but  is  due  to  an  error  in 
the  instrument  or  the  catalogue  and  correct  its  results  accordingly. 

(c)  When  circumpolar  stars  are  observed  both  above  and  below 
the  pole,  the  systematic  error  must  be  zero  at  the  pole. 

(d)  The  polar  correction  being  0,  and  the  equatorial  correction 
€,  it  is  assumed,  in  the  case  of  each  good  instrument  or  catalogue, 
that  the  error  varies  uniformly  between  these  limits. 

(e)  South  of  the  equator  the  error,  in  the  absence  of  any  means 
of  determining  it,  was  assumed  to  be  nearly  constant. 

The  positions  of  this  system  have  been  used  in  all  the  national 
Ephemerides  except  that  of  Germany  from  1901  to  the  present  time. 
They  were  introduced  into  the  American  Ephemeris  from  1900. 

4.  The  new  system  of  the  Berliner  Jahrbuch,  by  Auwers. 
During  the  years  1895-1903  Auwers  was  engaged  in  a  thorough 
reconstruction  of  the  A.G.  system,  resulting  in  a  new  funda- 
mental catalogue.  The  catalogue  itself  has  not  yet  appeared, 


§  191.]  EXISTING  FUNDAMENTAL  SYSTEMS  363 

but  the  corrections  to  the  A.G.  positions  and  proper  motions  of 
the  stars,  which  will  suffice  for  constructing  it,  are  found  in  the 
Astronomischen  Nachrichten,  vol.  164.  The  work  of  Auwers 
not  being  published  in  all  its  details,  a  description  of  his  methods 
cannot  yet  be  given. 

5.  The  new  Boss  system.  At  the  same  time  that  Auwers  was 
carrying  on  his  work,  Boss  was  also  constructing  a  new  funda- 
mental catalogue.  The  resulting  places  of  the  stars  of  the 
southern  hemisphere  appeared  in  1898,  and  will  be  found  in 
the  Astronomical  Journal,  vol.  xix.  The  positions  of  the 
northern  stars  were  completely  worked  out  in  1903,  and  the 
results  will  be  found  in  vol.  xxiii.  of  the  same  publication. 

The  question  of  the  systematic  differences  between  the  last 
three-named  systems  is  of  interest.  Practically,  they  may  be 
regarded  as  identical  so  far  as  the  Right  Ascensions  are  con- 
cerned. This  identity  arises  from  the  fact  that  all  three 
•catalogues  were  based  on  the  same  adopted  position  of  the 
equinox,  and  that  the  systematic  errors  of  observation  in  Right 
Ascension,  which  we  may  suppose  to  arise  from  the  diurnal 
changes  of  temperature,  are  largely  eliminated  in  the  course 
of  a  year's  work  with  a  good  instrument.  There  will  naturally 
be  small  differences  of  the  form  Aocg,  but  these  prove  not  to 
be  great  except  near  the  pole,  where  determinations  are 
necessarily  a  little  indefinite,  owing  to  uncertainty  as  to  the 
personal  equation  of  observers. 

In  the  case  of  the  declinations,  the  differences,  though  small, 
<ire  well  marked.  Near  the  pole  all  three  authorities  agree, 
«as  they  should,  because  all  systematic  errors  of  good  determina- 
tions are  small.  But,  from  20°  polar  distance  to  the  equator, 
Auwers  places  the  stars  a  little  farther  south  than  Newcomb, 
and  Boss  farther  south  than  Auwers.  Near  the  equator,  where 
the  difference  is  a  maximum,  the  mean  corrections  to  the 
-declinations  of  Newcomb,  as  found  by  the  two  authorities,  are :' 

Auwers,  Corr.=  -0"'13. 

Boss,  Corr.==  -0"'28. 

Whence  Boss  —  Auwers  =  —  0"'l  5. 


364  CATALOGUES  OF  THE   STARS  [§  191. 

These  differences  will  increase  very  slowly  owing  to  corre- 
sponding differences  in  the  proper  motions.  From  such  re- 
examination  as  the  author  has  been  able  to  give  to  the  subject, 
the  presumption  seems  to  be  that  his  system  does  really  require 
a  correction  in  the  direction  indicated  by  Boss  and  Auwers, 
and  the  probability  is  that  the  truth  lies  somewhere  between 
these  two  authorities.  The  difference  of  0"-15  between  them  is 
too  small  to  be  of  serious  import  for  the  present. 

In  connection  with  all  three  catalogues  are  given  tables  of 
systematic  corrections  of  the  form  already  described  for  the 
positions  of  all  the  principal  catalogues  of  stars. 


Section.  III.    Methods  of  Combining  Star  Catalogues. 

192.  Use  of  star-catalogues. 

We  have  shown  in  Chapters  XV.  and  XVI.  how,  when  the 
mean  position  of  a  star  at  some  epoch  is  given,  and  its  proper 
motion,  its  apparent  position  at  any  time  may  be  found.  We 
have  now  to  show  how  these  fundamental  elements  are  derived 
by  combining  the  data  given  in  various  star  catalogues. 

The  term  catalogue  of  precision  is  applied  to  those  catalogues 
of  which  the  purpose  is  to  give  precise  positions  of  the  stars.  The 
designation  is  used  in  contradistinction  to  the  lists  which  are 
intended  only  to  enable  the  stars  to  be  identified,  or  in  which 
precision  is  sacrificed  to  number. 

An  independent  catalogue  is  one  in  which  the  positions  are 
derived  solely  from  a  limited  number  of  observations  made 
at  some  one  observatory.  There  are  also  catalogues  which  give 
positions  of  stars  based  on  a  combination  of  the  work  of  various 
observatories,  with  a  view  of  deriving  as  accurate  results  as 
possible,  but  at  present  we  are  not  concerned  with  these. 
.  The  results  and  data  usually  given  in  a  catalogue  are  as- 
follows : 

1.  The  R.A.  and  Dec.  of  each  star,  as  derived  from  all  the 
observations,  and  referred  to  the  mean  equator  and  equinox 
of  some  convenient  epoch. 


§192.]  USE  OF  STAR-CATALOGUES  365 

2.  The  mean  date  of  all  the  observations  on  the  star. 

3.  The  number  of  observations  on  which  the  place  depends. 

4.  The  precessions  in  R.A.  and  Dec.  for  the  epoch  of  reference. 

5.  The  secular   variation  of  the    precessions.     In    most  inde- 
pendent catalogues,  this  refers  to  the  precession  alone,  not  to  the 
annual  variation. 

6.  The  proper  motion  of  each  star,  when  obtainable. 

It  should  be  added  that,  in  independent  catalogues,  the  proper 
motions  are  added  merely  for  the  convenience  of  the  astronomer 
using  the  work,  and  cannot  in  rigour  be  considered  as  belonging 
to  it,  because  they  cannot  be  based  on  the  same  observations 
as  the  positions  of  the  catalogue  itself. 

In  recent  catalogues  the  beginning  of  the  solar  year  is  adopted 
as  the  epoch  of  reference.  But,  in  former  years,  the  distinction 
of  the  civil  from  the  solar  year  was  little  attended  to,  and  the 
equinox  is  frequently  called  that  of  January  1,  although,  quite 
likely,  the  beginning  of  the  solar  year  was  actually  used. 

The  relation  of  the  epoch  of  reference  to  the  mean  date  of  the 
observations  must  be  understood.  If,  in  constructing  the  cata- 
logue, the  observed  positions  were  reduced  to  the  epoch  of 
reference  by  precession,  aberration,  and  nutation  alone,  without 
applying  a  correction  for  proper  motion,  the  given  position  will 
be  that  for  the  mean  date,  though  the  equator  and  equinox  will 
be  those  of  the  common  chosen  epoch.  This  is  theoretically  the 
best  course.  In  many  catalogues,  however,  the  proper  motion  for 
the  interval  between  the  date  and  the  epoch,  as  well  as  the 
precession,  is  applied  in  the  reduction,  with  a  view  of  giving 
the  actual  position  of  the  star  at  the  same  epoch  as  that  of 
the  equinox  of  reference.  The  user  of  the  catalogue  should 
always  know  which  system  is  adopted,  and  use  the  results 
accordingly. 

The  problem  of  deriving  the  position  of  a  star  for  any  date 
from  a  modern  catalogue  of  precision  requires  only  the  appli- 
cation of  formulae  and  methods  already  developed  in  the 
chapters  on  the  reduction  of  positions  of  the  fixed  stars.  The 
principal  question  left  open  will  be  whether  to  reduce  the  mean 
place  from  the  epoch  of  the  catalogue  to  the  required  epoch 


366  CATALOGUES  OF  THE   STARS  [§  192. 

by  means  of  the  precession,  proper  motion,  and  secular  variation 
found  in  the  catalogue,  or  by  the  trigonometric  method.  The 
choice  will  depend  on  the  length  of  the  interval  and  the  declina- 
tion of  the  star.  As  a  rough  and  ready  rule,  it  may  be  said  that, 
if  the  product  of  the  interval  in  years  by  the  secant  of  the 
declination  exceeds  40,  the  trigonometric  method  should  be 
adopted  ;  but,  if  less  than  40,  the  development  in  powers  of  t 
may  be  used,  if  it  is  found  more  convenient.  But  this  would 
prescribe  the  trigonometric  method  of  reducing  stars  within 
1°  30'  of  the  pole,  even  through  a  single  year,  where  it  would 
not  be  necessary.  The  limiting  value  of  the  product  (T—  T)  sec  S 
may,  therefore,  be  carried  up  to  50,  or  even  100  or  more,  near 
the  pole. 

In  ordinary  astronomical  practice,  the  position  of  a  star  found 
in  this  way  from  any  standard  catalogue,  or  in  any  of  the 
modern  independent  catalogues,  will  be  precise  enough  for 
general  use.  The  problem  we  have  to  consider  is  that  of 
deriving  the  position  and  proper  motion  of  a  star  with  the 
highest  attainable  precision  from  a  combination  of  all  the  in- 
dependent catalogues  in  which  it  is  found. 

193.  Preliminary  reductions. 

Having  found  the  star  in  any  catalogue,  certain  preliminary 
steps  will  be  required  to  reduce  the  data  to  the  required  form. 
These  are : 

1.  Possible  reduction  /or  proper  motion.  As  already  men- 
tioned, each  catalogue  has  two  epochs :  one  the  mean  epoch  of 
all  the  observations,  the  other  the  epoch  of  the  equinox  of 
reference.  It  should  be  understood  that  the  latter  epoch  really 
has  nothing  to  do  with  time,  because  it  merely  defines  the 
particular  system  of  coordinates  to  which  the  position  is  re- 
ferred. Time  enters  only  as  the  simplest  method  of  defining 
the  direction  of  the  fundamental  axes  of  reference;  the 
problems  growing  out  of  this  direction  are  purely  geometric. 

When  a  uniformly  varying  quantity  is  observed  at  several 
dates,  and  the  mean  of  all  the  results  taken,  this  mean  is  the 
most  probable  value  for  the  mean  date  of  all  the  observations, 


§  193.]  PKELIMINAEY  REDUCTIONS  367 

irrespective  of  the  rate  of  variation.  It  follows  that  the  ideally 
proper  position  to  give  in  a  catalogue  is  the  mean  of  the 
observed  positions  referred  to  such  equinox  of  reference  as  may 
have  been  selected. 

When  the  observed  position  is  reduced  from  the  mean  date 
of  the  observing  to  the  date  of  the  equinox  of  reference,  by 
applying  the  adopted  motion  during  the  interval,  the  position 
is  no  longer,  rigorously  speaking,  an  observed  one,  but  one  in 
which  observation  and  the  reduction  for  proper  motion  are 
combined.  It  follows  that,  if  the  computer  desires  to  follow 
a  rigorous  system,  he  should,  in  all  catalogues  where  the  re- 
duction in  question  is  made,  free  the  given  place  of  the  star 
from  this  correction.  Although  this  modification  is  seldom  of 
practical  importance  in  the  work,  the  habit  of  adopting  rigorous 
methods  in  astronomy  cannot  be  too  highly  recommended.  If 
the  reduction  is  not  made,  the  given  position  will  be  regarded 
as  if  observed  at  the  epoch  of  the  equinox  of  reference. 

2.  Systematic  corrections.     The  next  step   is   to   apply  such 
systematic  corrections  to  the  coordinates  of  the  star  as  may  be 
required  to  reduce  them  to  a  homogeneous  system.     The  method 
of  deriving  these   corrections   has   already   been  set  forth ;   but 
the  computer  will  rarely  have  to  do  this,  as  the  three  sets  of 
existing  tables   of  corrections    are    sufficiently   accurate  for   all 
practical  purposes.     In  making   a    choice   of  or  a  combination 
among  the  authorities,  it   might   be  a   good    practical    rule    to 
prefer  the  correction  of  the  smallest  absolute  amount,  because, 
as  already  pointed  out,  the  probability  is  that  such  a  correction 
will  be  too  large.     When  so  large  that  it  cannot  be  doubted, 
it  indicates  some  source   of  systematic  errors  in  the  catalogue 
which  should  diminish  the  weight  assigned,  and  therefore  the 
effect  of  the  correction  upon  the  final  result.     If  the  catalogue  is 
one  for  which  no  table  of  corrections  is  found,  one  may  easily  be 
constructed  by  the  methods  of  the  last  chapter. 

3.  Assignment  of  weights.     The  next  step  will  be  to  assign 
a  weight  to  the   catalogue    position   thus   corrected.     Were  all 
observations  completely  independent  determinations,  and  equally 
good,  the  weights  in  each  case  would  be  proportional  to  their 


368  CATALOGUES  OF  THE   STARS  [§  193. 

number.  But  each  instrument  has  peculiarities  of  its  own,  in 
virtue  of  which  the  determinations  of  any  one  star  with  it 
may  be  affected  by  a  constant  error,  which  will  be  less  the 
better  the  instrument.  Although  this  constant  error  may,  if 
considerable,  be  diminished  by  the  systematic  corrections,  it 
will  never  be  reduced  to  zero.  We  are,  therefore,  to  consider 
that  the  probable  error  e  of  a  position  taken  from  any  catalogue 
is  determined  by  the  equation 


€0  being  the  .probable  amount  of  the  constant  error,  and  el  that 
of  the  varying  accidental  error.  The  weights  are  then  taken 
so  as  to  be  inversely  proportional  to  e2,  the  general  form  being 


where  e  is  the  probable  error  chosen  to  correspond  to  the  unit 
of  weight,  and  n  the  number  of  observations. 

There  are,  of  course,  great  diversities  in  the  precision  of  the 
observations  on  which  various  catalogues  depend.  This  must 
be  taken  account  of  in  assigning  the  weights. 

Careful  investigations  of  the  data  on  which  the  various  cata- 
logues have  been  constructed  with  a  view  of  expressing  the 
weight  as  a  function  of  the  number  of  the  observations  have 
been  made  in  connection  with  the  three  fundamental  catalogues 
already  described.  It  will  hardly  be  worth  while  in  the  case  of 
any  existing  catalogue  for  a  computer  to  reinvestigate  this 
subject  for  himself.  He  can  either  adopt  one  of  the  latest  tables, 
or  combine  any  two  or  all  three  according  to  his  judgment. 

The  result  of  the  three  processes  will  be,  in  the  case  of  any 
one  catalogue,  that : 

At  a  certain  epoch  t  (that  of  the  mean  of  all  the  observations), 
the  right  Ascension  or  Declination  of  the  star,  referred  to  the 
mean  equator  and  equinox  of  an  epoch  T,  had  a  certain  value  a. 
or  <5,  and  that  this  value  is  entitled  to  a  certain  weight  w. 

The   star  being  found   in   as   many  good  catalogues  as  it  is 


§194.]  PRELIMINARY   REDUCTIONS  369 

thought  worth  while  to  use,  the  results  for  the  different  places 
of  the  stars  will  then  be  that : 


At  the  epochs  tv    t2,    t^, ...   tn 

For  the  equinoxes  of  Tv  T2,  T^...  Tn 

The  R.A.  or  Dec.  was  OLV  oc2,  oc3, ...  CLn 

With  the  respective  weights  wv  wz,  ws, ...  i.vn 


(a) 


194.  The  two  methods  of  combination. 

The  problem  now  is.  from  these  data  to  derive  the  most  likely 
mean  position  at  some  chosen  epoch  T0,  and  the  proper  motion. 
There  are  two  ways  of  doing  this,  of  which  the  first  is  simplest 
in  principle,  but  not  always  the  easiest  in  practice  : 

1.  Each  of  the  positions  OLV  a2,  ...may  be  separately  reduced 
to  the  mean  equinox  of  the  chosen  epoch  TQ  by  precession  alone, 
using  the  trigonometric  method  when  advisable.     We  shall  then 
have  a  series  of  values  of  a  which,  were  the  position  of  the  star 
•on  the  sphere  invariable,  and  the  catalogue  places  perfect,  should 
all  be  identical.     Differences  among  these  numbers  arise  from 
errors  of  the  catalogue  places,  and  from  the  proper  motion  of 
the  star.     The  best   position   and    proper    motion    can    then   be 
•determined  by  the  method  of  §§  39-41. 

2.  With   an   approximate  position  oc0   for   any   date   and   an 
^assumed  proper  motion  JULQ  the  position  of  the  star  may  be  com- 
puted for  the  several  epochs  Tv  T2)  etc.,  and  compared  with  the 
positions  given  in  or  derived  from  the  catalogue.     The  excess 
of  each  catalogue  position  over  the  computed  position  gives  a 
-correction  to  the  latter  for  the  mean  date  of  the  catalogue ;  and 
from  the  combination  of  all  these  corrections  the  most  likely  cor- 
rections to  oc0  and  //,0  may  be  derived  by  a  least-square  solution. 

It  will  be  seen  that  the  fundamental  difference  between  the 
two  methods  is  that,  in  using  the  first,  we  reduce  each  observed 
place  to  the  initial  epoch,  while  in  using  the  second  we  reduce  an 
assumed  place  for  the  initial  or  some  other  epoch  to  the  date 
of  each  observed  place. 

This  second  method  is  preferable  in  the  case  of  those  funda- 
mental and  other  stars  for  which  positions  for  various  dates 

N.S.A.  2A 


370  CATALOGUES   OF  THE   STARS  [§194, 

derived  from  a  single  value  each   of  oc0  and  /x0  are  available. 
Otherwise  method  1  is  preferable. 

195.  Development  of  first  method. 

The  epoch  T0  of  the  equator  and  equinox  to  which  the  positions 
are  to  be  reduced  must  first  be  decided  on.  If  a  general  cata- 
logue for  astronomical  researches  based  on  past  as  well  as  present 
observations  is  in  view,  the  most  convenient  epoch  will  probably 
be  1875,  as  this  was  in  extensive  use  during  the  last  quarter  of 
the  nineteenth  century,  and  is  that  to  which  the  catalogues  of  the 
Astronomische  Gesellschaft  are  reduced.  But,  if  the  positions- 
are  required  only  for  current  use,  it  will  be  better  to  choose 
1900,  or  even  some  later  epoch,  according  to  the  requirements. 

All  the  positions  are  then  to  be  reduced  from  their  several 
equinoxes  Tv  TZJ  etc.,  to  the  selected  equinox  TQ,  by  precession 
alone.  For  all  the  remoter  epochs  this  is  to  be  done  trigo- 
nometrically.  But  when  the  interval  of  reduction  is  short,  arid 
the  star  not  near  the  pole,  it  may  be  found  most  convenient  to 
use  the  annual  precessions  for  the  two  epochs,  or  that  for  the 
middle  epoch,  or  that  for  any  date  not  too  remote  from  the 
epoch,  combined  with  the  secular  variation.  When  the  latter 
is  used  the  proper  motion  should,  in  rigour,  be  omitted  in 
computing  it ;  but  commonly  the  effect  of  including  it  will  be- 
so  slight  that  its  retention  or  omission  will  be  unimportant. 

It  can  very  seldom  be  worth  while  to  compute  the  secular 
variations  for  this  express  purpose.  With  the  aid  of  the  tables 
given  in  Appendix  IV.  the  trigonometric  reduction  is  so  easy 
that  it  may  involve  less  labour  to  use  it,  even  for  an  interval  as 
short  as  ten  years,  than  it  will  to  compute  and  apply  the  annual 
precessions  and  secular  variations. 

When,  as  is  always  the  case  in  modern  catalogues,  the  pre- 
cessions for  the  date  of  the  catalogue  are  given,  these  may  be 
used,  care  being  taken  to  first  reduce  them  to  one  and  the  same 
standard  value  of  the  precessional  constant.*  Using  precessions, 
some  one  of  the  following  formulae  may  be  applied.  Put 


*  Tables  for  reducing  Struve's  precession  to  those  adopted  in  the  present  work 
are  found  in  Appendix  V. 


§  196.]  DEVELOPMENT  OF   FIRST  METHOD  371 

pt,  the  annual  precession  for  the  date  Ti  of  the  catalogue  to  be 

reduced  ; 

p0,  that  for  the  fundamental  epoch  TQ\ 
s,  the  secular  variation. 
Aoc^,  the  reduction  of  the  catalogue  position  to  the  adopted 

fundamental  equinox. 

Then, 


or 

In  such  a  case  as  this,  what  is  wanted  is  the  position  referred 
to  the  equinox  T0,  as  observed  at  the  epoch  Tit  without  the 
application  of  proper  motion  for  the  interval.  Hence,  if  the 
preceding  formulae  are  used,  the  precession  pQ  for  the  date  T0 
should  not  be  computed  with  the  actual  place  of  the  star  at  that 
epoch,  but  with  the  place  as  it  would  be  found  without  applying 
the  proper  motion  from  Ti  to  T0.  But  it  is  only  in  the  case  of 
exceptionally  large  proper  motions  that  attention  to  this  point  is 
necessary.  The  criterion  is  whether  the  change  in  the  position 
of  the  star  produced  by  the  proper  motion  during  the  interval  is 
large  enough  to  materially  affect  the  precession  p0. 

It  must  also  be  noted  that  the  formulae  cease  to  be  applicable 
when  the  star  is  so  near  the  pole  that  the  angle  S  for  the  interval 
TQ  —  TI  cannot  be  treated  as  infinitesimal.  The  trigonometric 
method  should  be  used  in  all  these  exceptional  cases. 

196.  Formation  and  solution  of  the  equations. 

Having  the  results  of  the  reductions  as  arranged  in  the  scheme 
(a),  §  193,  the  problem  is  to  find  the  position  and  proper  motion 
which  will  best  satisfy  the  observations.  We  may  in  all  but  the 
extremest  cases  regard  the  proper  motion  as  constant  when 
referred  to  the  pole  of  T0.  Even  in  the  exceptional  extreme  cases, 
we  may  proceed  on  the  supposition  of  uniform  proper  motion  if 
only  we  regard  the  result  as  the  value  of  a  uniformly  variable 
proper  motion  at  the  mean  epoch  of  all  the  observations.  Thus 
we  may  represent  each  reduced  R.A.  (or  Dec.)  as  giving  an 
equation  of  condition  of  the  form 

OL0-\-/uit=  reduced  oc. 


372  CATALOGUES  OF  THE   STARS  [§  196. 

Here  we  apply  the  method  developed  in  §  39  in  the  following  way : 

We  shall  designate  the  reduced  KA.'s  or  Decs,  by  04,  04, ...  oc^, 
it  being  noted  that  these  symbols  now  have  not  the  same 
meaning  as  in  (a),  being  all  reduced  to  one  equinox. 

The  solution  may  be  effected  by  arranging  the  several  results 
in  columns  in  tabular  form,  as  follows : 

Column  1 :  the  abbreviated  designation  of  the  several  star 
catalogues,  in  the  order  of  time. 

Column  2 :  the  several  values  of  tit  the  mean  date  of  the 
observing  for  each  catalogue.  It  will  not  be  necessary  to  write 
the  century  in  full,  and,  in  fact,  it  may  be  most  convenient  to 
write  down  instead  of  a  year  the  interval  in  years  before  or  after 
1850.  Then,  in  the  case  of  Bradley 's  catalogue,  the  earliest  of  all, 
we  should  have  t=  —0'95,  and  for  all  dates  before  1850 1  would  be 
negative.  In  this  column  it  will  ordinarily  be  most  convenient 
to  use  the  century  as  the  unit,  which  is  done  by  simply  putting  a 
decimal  point  before  the  tens  of  years. 

Column  3:  the  reduced  values  of  the  R.A.  oc^  (or  Declination) 
for  the  common  fundamental  epoch  T0.  In  all  ordinary  cases 
it  is  only  necessary  to  write  down  the  seconds  of  the  coordinate, 
the  hours  or  degrees  and  minutes  being  commonly  the  same 
for  all  catalogues. 

Column  4,  the  assigned  weights.  Practically  a  single  signifi- 
cant figure  will  be  enough  to  use  for  this  purpose,  or  two  for  the 
numbers  between  10  and  15.  That  is  to  say,  a  weight  of  54 
may  be  called  50,  one  of  56,  60,  etc.  While  it  is  true  that  the 
numerical  result  may  be  slightly  different,  the  difference  will 
never  be  more  than  a  small  fraction  of  the  uncertainty.  In  fact 
the  weight  is  always  under  any  circumstances  a  most  uncertain 
datum  with  which  to  deal. 

These  four  columns  contain  our  data  complete.  The  next  step 
is  to  multiply  the  columns  2  and  3,  t  and  a,  by  the  weights, 
writing  the  products  in  columns  5  and  6.  We  also  form  the 
product  wf2-  in  each  line,  which  we  may  do  by  multiplying  wt  by 
t,  and  which  goes  into  column  7,  and  wtoi,  which  goes  into 
column  8.  As  a  check  w  may  be  multiplied  mentally  by  the 
square  of  T. 


§  197.]    FORMATION  AND  SOLUTION  OF  THE  EQUATIONS      373 

The  sum  of  each  column  from  4  to  8  will  then  be  formed, 
giving  the  values  of  W,  [t],  [x],  or  [oc],  [tt]  and  [tx],  or  pa]. 

The  value  of  z  from  (63)  of  §  39  will  be  the  concluded  seconds 
of  the  coordinate  for  the  epoch  T0,  and  y  will  be  the  proper 
motion  for  the  mean  of  all  the  epochs,  referred  to  the  pole  of  T0. 
In  the  exceptional  case  of  large  proper  motions  of  stars  near  the 
pole,  it  will  be  necessary  to  make  a  reduction  for  the  changing 
values  of  the  proper  motions  even  when  referred  to  the  same 
equator  and  equinox.  The  theory  of  this  has  already  been  set 
forth. 

197.  Use  of  the  central  date. 

While  the  preceding  method  embodies  all  the  operations  really 
necessary  for  the  result,  there  will  be  a  certain  advantage  both 
in  symmetry  of  method  and  probable  freedom  from  error  by 
adopting  the  modification  developed  in  §40.  The  writer  believes 
that  the  additional  ease  and  symmetry  thus  secured  will  com- 
pensate for  the  slightly  increased  labour. 

By  this  method  we  do  riot  form  wt2  column  7,  but,  after  com- 
pleting column  6,  find  the  mean  date  of  all  the  observing,  or 
value  of  t0  by  the  equation 


This  mean  date  may  be  called  the  central  date.  It  has  the 
property  already  pointed  out,  that  the  most  likely  value  of  a  for 
that  date  is 

_[WCL] 
ao--^r 

independently  of  the  proper  motion  ;  and  also  at  this  epoch  the 
weight  of  the  position  derived  from  all  the  observing  is  a  maxi- 
mum, and  diminishes  symmetrically  with  the  time  before  and 
after  this  epoch. 

Having  found  the  central  epoch,  we  use  it  as  that  from 
which  t  is  counted,  following  the  method  of  §  40.  The  columns 
following  (6)  will  then  contain  the  successive  quantities 

Ftrf] 

^-^~!  =  T;  WT*;  WTCL. 


374 


CATALOGUES   OF  THE   STARS 


[§  197. 


Yet  more  perspicuity  will  be  secured  if,  instead  of  writing 
in  column  8,  we  subtract  the  weighted  mean  ot0  of  all  the 
a's  from  each  separate  oc,  calling  the  residual  r,  and  form  WTT. 
This  will  make  it  easy  to  see  the  relation  between  the  residuals 
and  proper  motions,  and  in  case  of  any  serious  divergence  or 
error  will  bring  it  out.  No  other  solution  of  an  equation  will 
then  be  necessary.  We  shall  have  for  the  proper  motion 

_[rr] 


Since  (Xo  is  the  definitive  coordinate  for  the  central  epoch,  the 
value  for  the  epoch  TQ  will  be 

CL0  +  im(T0-Cent.  ep.). 

The  proper  motion  ^t  will  be  that  for  the  central  date. 

The  following  form  of  computation  is  that  above  suggested. 
The  Catalogues  named  in  the  first  column  are  only  taken  as 
examples,  and  have  no  preference  over  others. 


1 

CATALOGUE. 

2 

Mean 
t 

3 

Sec. 
of  a 

4 
Wt. 

w 

5 

Wt 

6 

wa 

7 

Wt* 

orr 

8 
t^te 
or  r 

9 

WT 

10 

11 

icrr 

Brad.,  1755, 

-•95 

a, 

V, 

wltl 

wiai 

r, 

ri 

WIT, 

2 
WjTi 

Wj  TJ  rj 

Piazzi,  1800, 

-•50 

a2 

w, 

wzt, 

w,a. 

T2 

rt 

W2T, 

^rj 

^2  T2  r2 

Argel.,  1830, 

-•20 

a3 

w, 

^o3t3 

wsas 

r, 

rs 

»»n 

2 

^3  T3  ^3 

Pond,  1830, 

-•20 

a4 

W4 

w4t4 

W4a4 

n 

n 

™*r4 

2 

^4  T4  ^4 

Grh.,  1840, 

-•10 

«5 

™* 

™5t5 

w8aB 

Tg 

r5 

^r5 

^T? 

"5  ^5  ^5 

Pulk.,  1845, 

-•05 

a6 

W6 

iv6t6 

™6"6 

Tg 

r6 

™6T6 

2 

W6  ^6  ^6 

Grh.,  1872, 

+  •22 

*7 

w7 

w7t7 

«v% 

T7 

r7 

W7T7 

2 
W7T7 

^7  r7  ^7 

Pulk.,  1885, 

+  •35 

*8 

«t 

w8ts 

W8a8 

T8 

rs 

*Vsr8 

2 

^8r8r8 

Grh.,  1890, 

+  •40 

«9 

w9 

w9t9 

waa9 

T9 

r9 

W9  T9 

W9T! 

w»  T9  r9 

Mt.  Ham.,  1900, 

+  •50 

a10 

* 

«** 

»«* 

T10 

^0 

•M. 

2 

•y^ 

Sums, 

— 

W 

[«*] 

[wa] 

— 

— 

0 

W 

[rr] 

;§  198.]  USE   OF  THE   CENTRAL  DATE  375 

The  rigorous  application  of  the  above  method  will  be  impracti- 
cable when  a  catalogue  gives  only  one  coordinate,  or  when  the 
mean  date  of  observation  is  materially  different  for  the  two  coordi- 
nates, because  in  each  of  these  cases  we  have,  strictly  speaking,  no 
rigorous  determination  of  a  position  of  the  star  for  a  definite 
epoch,  which  is  necessary  for  the  reduction  to  T0.  Hence  the 
latter  cannot  be  made  except,  in  the  first  case,  by  using  an 
approximate  computed  value  of  the  missing  coordinate,  and,  in 
the  second  case,  by  reducing  the  two  observed  coordinates  to  the 
same  epoch  with  an  assumed  proper  motion.  In  all  ordinary  cases 
either  of  these  courses  may  be  taken  without  leading  to  error. 
In  exceptional  cases  it  is  preferable  to  adopt  the  second  method. 

198.  Method  of  correcting  provisional  data. 

'The  first  step  in  the  application  of  this  method  is  the  prepara- 
tion of  an  ephemeris  of  the  mean  RA.  and  Dec.  of  the  star  for 
the  several  catalogue  dates  Tv  T2,  etc.  In  the  case  of  the 
standard  stars,  such  an  ephemeris  may  easily  be  formed  from 
.data  given  in  general  catalogues  of  standard  stars.  Among 
tthose  which  may  be  used  for  this  purpose  are  the  catalogue  of 
1098  standard  clock  and  zodiacal  stars  found  in  Astronomical 
Papers  of  the  American  Ephemeris,  vol.  i.,  the  general  cata- 
logue of  standard  stars  in  vol.  ix.  of  the  same  papers,  and  the 
standard  catalogues  in  Publications  of  the  Astronomischen  Gesell- 
.schaft,  Nos.  xiv.  and  xvii. 

If,  from  any  of  these  catalogues,  we  compute  positions  of  a 
:star  for  the  epochs  of  observation,  each  excess  of  an  observed  over 
a  computed  position  will  give  a  correction  to  the  latter  for  the 
date  in  question. 

In  making  the  comparison,  attention  must,  of  course,  be  paid 
to  the  proper  motion  of  the  star  between  the  mean  date  of  all 
the  observations  used  in  forming  the  catalogue  place  and  the 
date  of  the  equinox  to  which  the  place  is  referred.  If  no  proper 
motion  is  applied  in  the  catalogue,  it  will  be  necessary  to  apply 
the  provisional  value  JU.Q.  If  that  actually  applied  is  materially 
different  from  the  provisional  value,  the  necessary  correction 
should  be  made. 


376  CATALOGUES  OF  THE   STARS  [§  198. 

In  any  case,  the  date  for  which  the  correction  holds  good  is 
not  that  to  which  the  catalogue  has  been  reduced,  but  the  mean, 
date  of  the  observations. 

The  results  of  the  comparisons  will  be  that  at  the  epochs 

the  provisional  place  of  the  star  seemed  to  require  the  corrections 

c  being,  in  each  case,  the  excess  of  the  R.A.  (or  Dec.)  of  the 
catalogue  over  the  provisional  one. 

In  nearly  all  cases  it  may  be  assumed  that  this  correction 
should  increase  uniformly  with  the  time.  We  may,  therefore,, 
proceed  as  in  method  1,  the  result  being,  instead  of  a  mean 
position  and  a  proper  motion,  corrections  to  the  assumed  mean 
position  and  proper  motion. 

If  we  compute  and  apply  this  correction  for  any  or  all  the 
dates  for  which  provisional  places  have  been  computed,  the 
result  will  be  the  corrected  places  for  the  same  dates. 

199.  Special  method  for  close  polar  stars. 

The  preceding  method  presupposes  that  the  angle  8  at  the- 
star  is  small.  When  such  is  not  the  case,  a  more  rigorous 
proceeding  is  necessary,  which  we  shall  now  briefly  indicate. 

The  elements  of  position  to  be  corrected  are : 

ao»    ^o>    Ma,    Ms,  (7) 

the  provisional  R.A.,  Dec.,  and  proper  motion  of  the  star  at  a 
certain  date  T0.  A.  coordinate  a  or  S  at  any  other  date  T  is 
a  function  of  these  four  quantities,  so  that  we  may  write 

a=/i(aO'  <V  Ma,  MsX 

$=f2(OL0>    <V    Me 

If,  following  the  general  method  found  in  §§  34,  35,  we  apply 
symbolic  corrections  Aoc0,  A(50,  etc.,  to  the  elements  (7),  the 
effect  of  these  corrections  upon  the  place  oc  will  be  given  by 

da.  „       .  den  A  ^    .  da,  A       .  da. 

I\M,S 

(9)> 
dS 


§199.]      SPECIAL  METHOD   FOR  CLOSE   POLAR   STARS 


377 


The  rigorous  method  is  to  compute  these  differential  co- 
efficients for  each  of  the  catalogue  dates,  and  put  their 
numerical  values  in  the  right-hand  member  of  (9),  while  in  the 
left-hand  member  we  put  for  Aoc  and  A$  the  excess  of  the 
catalogue  or  observed  over  the  provisional  coordinate.  Thus 
we  shall  have  a  conditional  equation  between  the  four  unknown 
quantities  Aoc0,  A(50,  Ayua,  and  Ajus,  and  by  solving  all  these 
equations  we  derive  the  values  of  the  corrections  to  the 
fundamental  elements. 

When  the  angle  S  is  small,  we  shall  have  very  nearly 

da.      dS 


_ 
~~ 


.(10) 


while  the  value  of  the  other  coefficients  will  be  so  small  that  the 
terms  multiplied  by  them  may  be  dropped.  We  should  then 
reproduce  the  equation,  the  solution  of  which  is  given  in  the 
scheme  of  §  197.  When  this  approximate  method  is  not  suffi- 
ciently exact,  we  must  seek  for  the  accurate  values  of  the 
differential  coefficients.  These  we  can  find  from  the  equations 
of  the  form  (8),  which  give  the  values  of  oc  and  S  in  terms  of 
oc0  and  8 


0  through  the  system  of  equations 


cos  8  sin  a  =  cos  ^  sin  a0 
cos  8  cos  a  =  cos  0  cos  S1  cos  a0  —  sin  0  sin  ^ 
si  n  3  =  sin  6  cos  <5X  cos  a0  -f-  cos  0  sin  Sl 


We  see  from  these  equations  that 

dot.       da,      doL     da, 


.(12) 


The  quantities    which   enter  into   the    equation    (11)  are   al] 
parts  of  the  spherical  triangle  $P0P  formed  by  the  star  and  the 


378  CATALOGUES   OF  THE   STARS  [§199.] 

two   mean    poles.     The   preceding  derivations  are    expressed  in 
terms  of  these  parts  as  follows : 

doL  cos  8Q  a 
-j —  =  —  -Qcos$ 
aoc0  cos  o 

doi  _sin  S 

d$0  ~  cos  8 

=  —  cos<50sin$ 


dS 


The  values  of  oc  and  8  in  (8)  are  found  by  trigonometric 
reduction  by  the  method  of  Chapter  X.  The  value  of  Act  and 
A3  in  (9)  are  the  corrections  to  this  reduced  place  given  by 
observations.  The  equations  (9)  are  then  solved  by  the  method 
of  least  squares  for  the  four  unknown  quantities  which  they 
contain,  resulting  in  corrections  to  the  adopted  provisional 
positions  and  proper  motions. 

This  method  will  become  more  and  more  necessary  as  more 
stars  near  the  pole  have  to  be  investigated,  and  as  the  period 
over  which  observations  extend  is  lengthened.  An  application 
of  it  to  the  four  north  polar  stars  most  used  will  be  found 
in  volume  viii.  of  the  Astronomical  Papers  of  the  American 
Ephemeris. 


NOTES  AND  REFERENCES. 

THE  mass  of  astronomical  literature  relating  to  positions  of  the  fixed  stars 
is  so  great  that  it  is  not  possible,  in  the  present  connection,  to  do  more  than 
cite  the  principal  independent  catalogues  of  stars,  and  offer  some  suggestions 
as  to  the  literature  of  the  subject.  From  what  has  already  been  said  of  the 
history  of  the  subject  it  will  be  seen  that  the  determination  of  positions  of 
the  fixed  stars  by  meridian  observations  has  formed  a  large  fraction  of  the 
work  of  the  leading  observatories  since  1750.  The  instruments,  the  system 
of  observation,  and  the  methods  of  reduction  and  combination  have  been  so 
frequently  imperfect  that  the  question  what  results  are  worth  using  often 
becomes  one  of  much  difficulty,  the  decision  of  which  must  be  left  to  the 
investigator  himself.  To  this  diversity  of  material  must  be  added  lack  of 
continuity  in  observations  and  in  systematic  forms  and  methods  of  publication. 


NOTES   AND    REFERENCES  379 

Since  the  middle  of  the  last  century,  following  the  example  of  Greenwich, 
it  has  been  quite  usual  for  observatories  making  regular  meridian  observa- 
tions to  publish  in  each  annual  volume  the  mean  positions,  for  the  beginning 
of  the  year,  of  all  the  stars  observed  on  the  meridian  during  the  year.  At 
intervals  of  a  few  years  these  annual  positions  have  been  generally,  but 
not  always,  combined  into  a  single  catalogue,  reduced  to  some  convenient 
equinox  near  the  mean  of  the  times  of  observation.  But  there  are  still 
several  series  of  observations,  some  of  which  are  probably  as  good  as  any 
made  during  their  time,  which,  although  published,  have  never  been  com- 
pletely reduced.  The  question  whether  it  would  be  profitable  to  utilize  such 
observations  is  one  that  frequently  arises,  but  has  to  be  postponed  for  want 
of  the  means  necessary  to  effect  the  reduction. 

Besides  the  independent  volumes  issued  by  observatories,  the  volumes  of 
the  Astronomische  Nachrichten,  the  number  of  which  will,  before  many  years, 
pass  the  200  mark,  contain  a  vast  amount  of  material  of  every  kind  relating 
to  the  subject,  which  should  be  accessible  to  the  investigator  who  wishes  to 
have  all  the  aids  which  may  possibly  be  useful  in  his  work.  Discussions 
relating  to  the  positions  of  stars  are  also  found  in  the  Astronomical  Journal, 
which  has  now  reached  its  25th  volume. 

Of  material  contained  in  these  and  other  serials  it  will  be  necessary  to  cite 
only  that  most  uniformly  essential,  namely,  the  systematic  corrections  to 
various  catalogues,  and  the  weights  to  be  assigned  to  the  given  positions  as 
a  function  of  the  number  of  observations  on  which  each  result  depends. 

Boss's  system  of  corrections  is  found  in  Astronomical  Journal,  vol.  xxiii., 
pp.  191-211. 

Auwers'  system  of  reductions  is  found  in  Astronomische  Abhandlungen  als 
Erganzungshefte  zu  den  Astronomischen  Nachrichten ;  Nr.  7,  Tafeln  zur  Reduc- 
tion von  Sterncatalogen  auf  das  System  des  Fundamentalcatalogs  des  Berliner 
Jahrbuchs. 

Newcomb's  system  of  corrections  is  found  in  Astronomical  Papers  of  the 
American  Ephemeris,  vol.  viii.,  chapter  iv. 

The  differences  between  the  reductions  given  by  these  different  authors 
arise  not  only  from  the  differences  of  the  fundamental  systems,  but  from 
differences  in  the  principles  on  which  the  corrections  were  derived.  The 
principal  difference  in  principle  is  that  in  forming  his  system  Newcomb 
required  more  evidence  that  a  systematic  difference  was  necessary  than  did 
either  Auwers  or  Boss. 

Auwers'  tables  of  weights  are  found  in  the  Ast.  Nach.,  vol.  151,  S.  225-274, 
under  the  title  :  Gewichstafeln  fur  Sterncataloge. 

The  assignment  of  weights  is  of  necessity  largely  a  matter  of  judgment, 
based  on  what  is  known  of  the  methods  of  making  the  observations  and 
constructing  the  catalogue.  Marked  diversity  in  the  different  systems  is 
therefore  to  be  expected. 


380  CATALOGUES  OF  THE   STARS 


LIST  OF   INDEPENDENT  STAR  CATALOGUES. 

THE  following  is  a  list  of  the  principal  independent  catalogues  of  precision 
which  may  be  available  in  investigating  positions  and  proper  motions  of 
stars.  The  term  "  catalogue  of  precision  "  is  used  in  a  somewhat  broad  sense, 
including  all  catalogues  which  were  intended  to  be  more  than  simple  lists  of 
stars.  Many  are  no  doubt  admitted  which,  on  critical  examination,  will  be 
found  below  others  that  have  been  excluded. 

From  the  list  are  also  omitted  observations  of  stars  in  zones,  and  catalogues 
constructed  from  them.  The  most  important  of  these  are  the  zones  observed 
by  Bessel  and  by  Argelander,  well-known  catalogues  from  which  have  been 
published  by  Weisse  and  by  Oeltzen.  Annual  catalogues  are  also  excluded, 
whether  they  have  been  combined  or  not. 

Another  class  excluded  is  that  in  which  the  given  positions  are  not 
independent,  but  are  derived  by  a  combination  of  other  observations  than 
those  made  especially  for  the  catalogue  in  question. 

In  tabulating  and  comparing  the  results  derived  from  different  catalogues 
it  is  necessary  to  have  the  briefest  distinctive  designation  of  each.  This  is 
commonly  either  the  abbreviated  name  of  the  observatory,  or  the  name  of 
the  author,  followed  by  the  date  of  the  catalogue.  For  the  latter  is  chosen 
sometimes  the  equinox  of  reference  and  sometimes  the  mean  date  of  the 
observations,  commonly  the  former.  In  the  list  which  follows  it  is  the  name 
of  the  observatory  or  place  of  observation  which  is  generally  given.  Boss 
introduced  the  system  of  abbreviating  the  name  of  the  observatory  to  its 
first  and  last  letters  which  is  convenient  in  writing  but  not  always  suffi- 
ciently explicit.  Auwers  uses  sometimes  the  name  of  the  observatory  and 
sometimes  that  of  the  author  of  the  catalogue. 

CATALOGUES  MADE  AT  NORTHERN  OBSERVATORIES. 

THE  first  complete  reduction  and  discussion  of  Bradley's  observations  was 
made  by  Bessel  and  published  in  1818  under  the  title  : 

Fundamenta  Astronomiae  pro  anno  MDCCL  V  deduct  a  ex  observationibus 
viri  incomparabilis  JAMES  BRADLEY.  In  Specula  Astronomica  Grenovicensi 
per  Annos  1750-1762  Institutis.  Auctore  FRIDERICO  WILHELMO  BESSEL. 
Regiomonti,  1818. 

This  work  is  now  superseded  by  that  of  Auwers  of  which  the  designation 
and  title  are  : 

AUWERS-BRADLEY,  1755. — Neue  Reduction  der  Bradley' schen  Beobachtungeny 
aus  den  Jahren  1750-1762,  von  ARTHUR  AUWERS.  3  volumes,  St.  Petersburg, 
1882-1903. 

The  catalogue  is  found  in  the  third  volume.  The  first  volume  contains  a 
valuable  discussion  and  comparison  of  the  observations,  which  will  serve  as 
an  excellent  model  to  the  astronomical  student  desiring  to  perfect  himself  in 


CATALOGUES  MADE  AT  NORTHERN  OBSERVATORIES     381 

methods  of  discussion.  The  R.A.'s  of  Auwers  leave  no  question  open  which 
it  would  be  profitable  to  discuss.  But  such  is  not  the  case  with  the  declina- 
tions, his  work  on  which  has  been  examined  by  the  present  author  as  well 
as  by  Boss.* 

AUWERS- MAYER,  1755. —  Tobias  Mayer's  Sternverzeichniss  nach  den  Beobacht- 
nngen  aufder  Go'ttinger  Sternwarte  in  den  Jahren  1756-1760.  Neu  Bearbeitet 
von  ARTHUR  AUWERS.  Leipzig,  1894. 

This  work  is  based  on  observations  by  Tobias  Mayer,  made  shortly  after 
the  epoch  of  Bradley,  with  whose  work  it  favourably  compares.  The 
number  of  stars  is,  however,  rather  small. 

PIAZZI,  1880. — Praecipuarum  Stellarum  inerrantium  Positiones  mediae  ex 
vbservationibus,  1792-1813.  Folio,  Panormi,  1814. 

This  catalogue  when  constructed  was  vastly  superior  to  any  that  preceded 
it,  and  is  still  of  value  in  determining  proper  motions.  But  it  is  now  far 
behind  modern  requirements.  It  is  being  reconstructed  by  Dr.  Herman  S. 
Davis,  under  the  auspices  of  the  Carnegie  Institution.  Until  this  work  is 
•completed  and  published  it  is  scarcely  worth  while  to  make  use  of  the 
catalogue  except  for  stars  not  observed  by  Bradley. 

GROOMBRIDGE,  1810. — A  Catalogue  of  Circumpolar  Stars,  deduced  from  obser- 
vations of  Stephen  Groombridge,  Esq.  Reduced  to  January  1,  1810.  Edited 
by  GEORGE  BIDDELL  AIRY,  ESQ.,  A.M.,  Astronomer  Royal.  London,  1838. 

The  observations  on  which  this  catalogue  is  based  were  made  by  an 
enthusiastic  amateur  at  Blackheath,  and  are  valuable  from  their  early  date, 
.and  the  number  of  circumpolar  stars  included.  The  above  cited  publication 
by  Airy  has  been  the  only  one  hitherto  available,  but  a  re-reduction  has 
recently  been  completed  at  the  Greenwich  Observatory,  and  the  catalogue 
based  upon  it  is  entitled — 

New  Reduction  of  Groombridge's  Circumpolar  Catalogue.  By  FRANK  W. 
DYSON  and  WILLIAM  G.  THACKERAY  under  the  direction  of  SIR  WILLIAM 
H.  M.  CHRISTIE.  London  Admiralty,  1905. 

The  principal  defect  in  Groombridge's  observations  is  that  very  few 
•observations  were  made  below  the  pole,  and  in  consequence  the  error  of  his 
instrument  in  azimuth  cannot  be  fixed  with  all  desirable  certainty. 

POND- AUWERS,  l&lb.—  Mittlere  Oerte  von  570  Sternen  .  .  .  aus  den  unter 
Direction  von  Pond,  1811-1819,  angestellen  Beobachtungen.  Von  A.  AUWERS. 
Berlin  Akademie,  1902. 

KONIGSBERG,  1820. — Neue  Reduction  der  Konigsberger  Declinationen  1820, 
von  W.  DOLLEN.  Found  in  Recueil  de  Memoires  presented  a  V Academic  des 
Sciences  par  les  Astronomes  de  Poulkova,  vol.  i.,  St.  Petersburg,  1853. 

This  reduction  includes  only  about  60  fundamental  stars,  to  the  determina- 
tion of  which  Bessel  devoted  special  attention. 


*  Astronomical  Papers  of  the  American  Ephemeris,  viii.,  p.   194;   Ast.  Jour., 
vol.  xxiii. 


382  CATALOGUES   OF  THE   STARS 

DORPAT,  1830. — Struve's  Fundamental  Catalogue  for  this  epoch  is  found 
in  his  Stellarum  Fixarum  Positiones  Mediae  Epocha  1830.  Auctore  F.  G.  W. 
STRUVE.  Petropoli,  1852. 

ARGELANDER,  1830. — DLX  Stellarum  Fixarum  Positiones  Mediae  Ineunte 
Anno  1830.  Helsingforsiae,  1835. 

This  catalogue  of  560  stars,  almost  all  of  the  brighter  class,  including 
especially  the  fundamental  stars,  is  based  on  Argelander's  observations  at 
Abo  before  his  removal  to  Bonn. 

POND,  1830. — A  Catalogue  of  1,1 12  Stars,  reduced  from  observations  made  at 
the  Royal  Observatory  at  Greenwich, from  the  years  1815  to  1833.  London,  1833. 

The  observations  on  which  this  catalogue  is  based  are  probably  good  when 
measured  by  the  standard  of  the  period.  Their  combination  in  the  catalogue 
is,  however,  not  carried  out  in  the  best  way,  and  the  re-reduction  and  recom- 
bination of  the  whole  is  to  be  desired.  This  is  partly  done  by  Auwers  in 
the  work  Pond,  1815,  above  cited.  The  result  of  Chandler's  discussion 
of  the  standard  declinations  is  found  in  Ast.  Jour.,  xiv.  A  correction  to  the 
catalogue  declinations  on  account  of  the  refractions  is  tabulated  by  Auwers  in 
Ast.  Nach.,  vol.  134,  col.  52. 

CAMBRIDGE,  1830. — The  First  Cambridge  Catalogue  of  726  Stars,  deduced 
from  the  Observations  made  at  the  Cambridge  Observatory,  from  1828  to  1835  ,- 
reduced  to  January  1,  1830,  by  GEORGE  BIDDELL  AIRY,  ESQ.,  Astronomer 
Royal,  etc. 

This  work  is  extracted  from  Memoirs  of  the  Royal  Astronomical  Society \ 
vol.  xi.,  pages  21  to  45,  London,  1840. 

The  probable  errors  of  this  catalogue  are  larger  than  would  have  been- 
anticipated  in  a  work  by  Airy.  It  seems  probable  that  a  defective  system 
of  reduction  and  combination  has  detracted  from  the  precision  of  the 
results.  If  so,  a  re-reduction  of  the  original  observations  is  desirable. 

KONIGSBERG,  1835. — Beobachtungen  von  Zodiacalsternen  am  Reichenbach- 
schen  Meridiankreise,  in  Astronomische  Beobachtungen  auf  der  Koniglichen 
Universitdts-Sternwarte  zu  Konigsberg,  von  DR.  EDTJARD  LUTHER.  Band  xxxvii., 
Zweiter  Theil.  Konigsberg,  1886. 

RUMKER,  1836. — Mittlere  Oerter  von  12000  Fix-Sternen,  von  CARL  RUMKER. 
Hamburg,  1852. 

EDINBURGH,  1840. — This  catalogue  is  based  on  observations  between  1836 
and  1845. 

ARMAGH  (ROBINSON),  1840. — Places  of  5,345  Stars  observed  from  1828  to 
1854,  by  REV.  T.  R.  ROBINSON.  Dublin,  1859. 

GILLISS,  1840. — Astronomical  Observations  made  at  the  Naval  Observatory 
Washington,  by  LIEUT.  J.  M.  GILLISS,  U.S.N.  Washington,  1846. 

This  work  contains  a  catalogue  of  1,248  stars,  mostly  zodiacal  and 
equatorial,  observed  in  connection  with  moon-culminations  between  1838- 
and  1842.  Only  the  R.A.'s  are  independent,  the  Decs,  being  taken  from 
the  B.A.  catalogue. 


CATALOGUES  MADE  AT  NORTHERN  OBSERVATORIES     383 

OXFORD,  1845,  1860,  and  1890.—  The  Radcliffe  Catalogues  of  Stars. 

The  old  meridian  instrument  at  the  Radcliffe  Observatory,  Oxford,  with 
which  these  observations  were  made,  was  of  inferior  construction,  and,  in 
consequence,  its  results  require  systematic  corrections,  varying  rapidly  with 
the  position.  The  introductions  to  the  annual  volumes  of  the  Radcliffe 
observations,  recent  papers  in  the  Monthly  Notices,  R.A.S.,  and  comparisons 
in  Ast.  Papers  of  the  American  Ephemeris,  viii.,  166-167,  should  be  consulted. 

CARRINGTON,  1855. — Catalogue  of  3,735  Circumpolar  Stars  observed  at  the 
Red  Hill  Observatory,  1854-56,  by  RICHARD  C.  CARRINGTON. 

The  stars  of  this  catalogue  are  all  situated  within  9°  of  the  pole.  The 
instrument  was  probably  not  of  the  best ;  but  the  catalogue  may  be  re- 
garded as  one  of  precision  and,  for  the  region  which  it  covers,  the  most 
complete  made  up  to  that  time. 

GREENWICH,  1855  to  1890. — Since  1836,  when  Airy  took  charge  of  the 
Greenwich  Observatory,  catalogues  based  on  the  observations  through, 
periods  ranging  from  six  to  ten  years  have  appeared  as  follows  : 

Epoch  of  Reference.  Years. 

1840  -         Years  of  observation,  1836-1841. 

1845  „  „  1842-1847. 

1850  „  „  1848-1853. 

1860  „  „  1854-1860. 

1864  -             „  „  1861-1867. 

1872  „  „  1868-1876. 

1880  „  „  1877-1886. 

1890  -             „  „  1887-1896. 

POULKOVA,  1845  to  1892. — The  Poulkova  standard  catalogues  have  ap- 
peared in  various  volumes  of  the  series — Observations  de  Pulkowa,  publiees 
par  OTTO  STRUVE.  Directeur,  etc.,  and  Publications  de  VObservatoire  Central 
Nicolas — and  also  independently.  In  some  cases  a  revised  edition  of  the 
catalogue,  which  should  be  used  instead  of  the  original,  has  been  issued. 
The  standard  catalogues,  in  some  of  which  the  R.A.'s  and  Decs,  are  given 
in  separate  publications,  are  found  in  the  following  volumes  : 

For  the  Epoch  1845  in  volumes  i.  and  iv. 

„        1865          „  xii. 

„  „        1885  in  serie  ii.,  vol.  i. 

„  „        1892         „       „    vols.  viii.-ix. 

A  corrected  list  of  the  standard  declinations  for  1845  is  published  as  a 
supplement  to  volume  iv. 

Poulkova  catalogues,  embracing  a  larger  number  of  stars,  are  cited  in  their 
chronological  order. 

POULKOVA,  1855. — Positions  moyennes  de  3542  e'toiles  de'termine'es  a  I'aide  du 
Cercle  Me'ridien  de  Poulkova  dans  les  annexes  1840-1869.  Observations  de 
Poulkova,  Vol.  VIII. 


384  CATALOGUES   OF  THE   STAES 

In  this  catalogue  an  error  was  made  by  the  computers  in  applying  the 
correction  for  errors  of  graduation  of  the  meridian  circle.  It  happens, 
however,  that,  as  the  corrections  in  question  vary  slowly  and  regularly  from 
one  declination  to  another,  and  as  all  declinations  of  the  stars  were  reduced 
to  the  standard  of  the  vertical  circle,  the  final  effect  of  the  error  upon  the 
positions  as  given  in  the  catalogue  is  unimportant.  The  subject  is,  however, 
discussed  very  fully  in  a  paper  by  Backlund,  found  in  the  St.  Petersburg 
Memoirs,  series  vii.,  volume  xxxvi.,  St.  Petersburg,  1888. 

PARIS,  1845,  1860,  and  1875.-- Catalogue  de  I' Observatoire  de  Paris.  JStoile* 
observers  aux  instruments  meridiens.  4  volumes,  4to,  1887-1902. 

This  catalogue  is  valuable  for  the  great  number  of  faint  stars  of  which  it 
gives  modern  positions. 

YARNALL,  1860. — Catalogue  of  Stars  observed  at  the  U.S.  Naval  Observatory 
during  the  years  1845-1877. 

Three  editions  of  this  catalogue  have  appeared,  the  last  being  thoroughly 
revised  by  Professor  Edgar  Frisby,  U.S.N.  The  work  labours  under  the 
disadvantage  of  including  two  distinct  series  of  observations ;  the  one 
beginning  in  1845  and  coming  nearly  to  a  stand-still  during  the  years 
1850-1860  ;  the  other  beginning  in  the  year  1861.  The  condition  of  the 
instruments  and  the  method  of  using  them  changed  so  much  during  this 
time  that  the  catalogue  as  a  whole  may  be  considered  as  a  combination  of 
two,  the  results  of  which  require  different  systematic  corrections. 

HARVARD,  1865. — Annals  of  Harvard  College  Observatory,  vol.  iv. 

This  catalogue  contains  R.A.'s  of  506  stars,  without  declinations. 

LEIDEN,  1870. — Declinations  of  202  Fundamental  Stars,  Annalen  der 
Sternwarte  in  Leiden,  Band  ii.,  p.  125,  and  Ast.  Nach.,  Ixxx.,  S.  94. 

GLASGOW,  1870. — Catalogue  q/"6415  Stars  deduced  from  observations  made  at 
-ihe  Glasgow  University  Observatory.  By  ROBERT  GRANT.  Glasgow,  University 
Press,  1883. 

HARVARD,  1875. — Catalogue  of  1213  Stars,  observed  during  the  years  1870- 
1879  with  the  Meridian  Circle  of  Harvard  College  Observatory,  by  WILLIAM  A. 
ROGERS  ;  Harvard  Annals,  volume  xv.,  part  i. 

WASHINGTON,  1875. — The  Second  Washington  Catalogue  of  Stars  from  ob- 
servations with  the  transit  circle  at  the  U.S.  Naval  Observatory  from  1866  to 
1891,  by  PROFESSOR  J.  R.  EASTMANN,  U.S.N.* 

POULKOVA,  1875. — Catalog  von  5634  Sternen  fur  die  Epoche  1875  aus  den 
Beobachtungen  am  Pulkowaer  Meridiankreise  wdhrend  der  Jahre  1874-1880,  von 
H.  ROMBERG.  Supplement  III.  aux  Observations  de  Poulkova,  St.  Petersboui  g, 
1891. 

BERLIN,  1875. — Ableitung  der  Rectascensionen  der  Sterne  des  Fundamental- 


*  It  should  be  noted  that  the  systematic  corrections  found  in  the  first  page  of 
Ast,  Papers  of  the  Am.  Eph.,  Vol.  VIII.,  to  the  declinations'1  of  this  catalogue 
are  not  applicable  to  the  printed  declinations,  but  only  to  an  unpublished  original. 


CATALOGUES  MADE  AT  NOETHEEN  OBSEEVATOEIES     385 

Cataloges  der  Astronomischen  Oesellschaft  aus  den  von  H.  Romberg  in  den 
Jahren  1869-1873,  angestellten  Beobachtungen  von  DR.  A.  MARCUSB. 
Beobachtungs- Ergebnisse  der  Koniglichen  Sternwarte  zu  Berlin.  Heft  No.  4. 
Berlin,  1888. 

IBID. — Resultate  von  Beobachtungen  von  521  Bradley'schen  Sternen,  am 
grossen  Berliner  Meridiankreise,  von  DR.  E.  BECKER.  Berlin  Observations, 
1881. 

ARMAGH,  1875.—  Second  Armagh  Catalogue  0/3300  Stars  for  the  epoch  1875, 
by  T.  E.  EOBINSON  and  J.  L.  E.  DREYER. 

ASTRONOMISCHE  GESELLSCHAFT,  1875,—Katalog  der  Astronomischen  Gesell- 
schaft,  Erste  Abtheilung,  +80°  bis  -2°.  In  fifteen  Parts,  of  which  Part  II. 
(70°  to  75°)  has  not  been  published.  Leipzig,  1890-1902. 

IBID.;  Zioeite  Abtheilung  :— Pt.  II.,  -6°  bis  -10°.  Leipzig,  1904-,  is 
the  only  section  which  has  yet  appeared. 

OXFORD,  1890.— Catalogue  of  6,424  Stars  for  the  Epoch  1890,  formed  from 
observations  made  at  the  Radcliffe  Observatory,  Oxford,  during  the  years 
1880  to  1893,  by  E.  J.  STONE. 

BERLIN,  1890. — Ergebnisse  der  1886-1891  am  grossen  Meridiankreise  der 
Berliner  Sternwarte  angestellen  Beobachtungen  der  Jahrbuchsterne,  von  F. 
KTJSTNER.  Astronomische  Nachrichten,  vol.  cxlii.,  S.  113-134. 

MADISON,  1890. — Publications  of  the  Washburn  Observatory  of  the  University 
of  Wisconsin,  vol.  viii.  Meridian  Circle  Observations,  1887-1892.  Madison, 
Wis.,  1893. 

GLASGOW,  1890. — Second  Glasgow  Catalogue  q/2156  Stars  from  observations 
made  during  the  years  1886-1892.  By  EGBERT  GRANT.  Glasgow,  University 
Press,  1892. 

MUNICH,  1892. —  Untersuchungen  iiber  die  astronomische  Refraction  mit  einer 
Bestimmung  der  Polhb'he  von  Munch  en  und  ihrer  Schwankungen  von  November 
1891  bis  October  1893  und  einem  Katalog  der  absoluten  Declinationen  von  116 
Fundamental- Sternen,  von  DR.  JULIUS  BAUSCHINGER.  Miincheii,  1896. 

POULKOVA,  1895. — Catalog  von  781  Zodiacalsternen  fur  Aequinoctium  und 
Epoch  1895.0,  von  M.  DITSCHENKO  und  J.  SEGBOTH.  St.  Petersbourg,  1903. 

MOUNT  HAMILTON,  1895. — Observations  upon  selected  Stars  of  the  Astrono- 
mische Gesellschaft  Catalogue  made  with  the  meridian  circle  of  the  Lick 
Observatory  by  MR.  E.  H.  TUCKER,  during  the  years  1894-95.  Ast.  Jour. 
vol.  xvii.,  No.  408.  Publications  of  the  Lick  Observatory,  vol.  iv.,  302. 

BERLIN  (BATTERMANN),  1895. — Beobachtungs- Ergebnisse  der  koniglichen 
Sternwarte  zu  Berlin.  Heft  8.  Berlin,  1899. 

BERLIN  (BATTERMANN),  1900,  Ibid.    Heft  10. 

MOUNT  HAMILTON,  1900. — Results  of  Observations  of  Circumpolar  Stars, 
Zodiacal  Stars,  and  Southern  Stars  ofPiazzi.  Publications  of  Lick  Observatory, 
vol.  vi. 

Auwers  applies  large  systematic  corrections  to  the  declinations  of  the 
southern  stars  in  this  catalogue,  which  are  probably  necessary  on  account  of 
N.S.A.  2  B 


386  CATALOGUES  OF  THE   STABS 

the  usual  tables  of  refraction  not  being  correct  for  an  altitude  of  1300  metres 
above  sea  level. 

CINCINNATI,  1890,  1895,  and  1900.— Publications  of  the  Cincinnati  Obser- 
vatory, Nos.  13,  14,  and  15,  by  JERMAIN  G.  PORTER,  Director.  These  cata- 
logues give  observed  positions  of  2000,  2030,  and  4280  stars  respectively. 


CATALOGUES   FROM  TEOPICAL  AND   SOUTHERN 
OBSERVATORIES. 

LACAILLE,  1750. — A  Catalogue  of  9766  Stars  in  the  Southern  Hemisphere 
from  the  observations  of  the  Abbe"  de  Lacaille  made  at  the  Cape  of  Good  Hope, 
in  the  years  1751-1752.  By  FRANCIS  BAILT,  ESQ.  London,  1847. 

The  origin  of  this  catalogue  is  mentioned  in  the  preceding  chapter. 
From  its  very  nature  it  cannot  be  regarded  as  a  catalogue  of  precision,  but 
it  is  cited  because  its  positions  may  be  useful  in  the  case  of  stars  not  found 
in  other  catalogues. 

PARAMATTA,  1 825.— Catalogue  of  7385  Stars,  chiefly  in  the  southern  hemi- 
sphere, from  the  observations  made  in  1822-26  at  the  Observatory  at  Paramatta, 
New  South  Wales,  founded  by  SIR  THOMAS  MACDOUGALL  BRISBANE  ;  the 
Catalogue  by  MR.  WILLIAM  RICHARDSON.  Lou  don,  1835. 

The  observations  on  which  this  catalogue  was  based  were  made  with  the 
transit  instrument  and  mural  circle  ;  a  few  of  them  by  Sir  Thomas  Brisbane 
himself,  but  mostly  by  Mr.  Charles  Riimker,  later  of  Hamburg,  and  Mr. 
Dunlop.  The  work  is  of  importance  as  being  the  first  catalogue  of  precision 
embracing  stars  too  far  south  to  be  visible  in  Europe.  So  far  as  the  writer 
is  aware,  the  precision  of  the  results  has  never  been  tested  by  modern 
methods. 

FALLOWS,  1830. — A  Catalogue  o/425  Stars  observed  during  the  years  1829-31 
at  the  Cape  Observatoi'y,  reduced  and  published  by  G.  B.  AIRY.  Memoirs  of  the 
Royal  Astronomical  Society,  vol.  xix. 

ST.  HELENA  (JOHNSON),  1830. — A  Catalogue  of  606  Principal  Fixed  Stars 
in  the  Southern  Hemisphere,  deduced  from  observations  at  the  Observatory,  St. 
Helena,  from  November,  1829,  to  April,  1833,  by  MANUEL  J.  JOHNSON. 
London,  1835. 

CAPE  (HENDERSON),  1833. — Thos.  Henderson  on  the  Declinations  of  the 
Principal  Fixed  Stars,  deduced  from  observations  made  at  the  Observatory, 
Cape  of  Good  Hope,  in  the  years  1832  and  1833.  Memoirs  of  the  Royal 
Astronomical  Society,  vol.  x. 

MADRAS  (TAYLOR),  1835. — Taylor's  General  Catalogue  of  Stars  from  observa- 
tions made  at  the  Madras  Observatory  during  the  years  1831-1842.  Revised  and 
edited  by  A.  M.  W.  DOWNING,  ESQ.  Edinburgh,  1901. 

This  catalogue,  based  on  the  work  of  an  industrious  observer,  is  of  decided 
value,  but  suffers  from  the  imperfection  of  the  instruments  with  which  the 
observations  were  made.  The  R.A.'s  especially  are  affected  by  a  systematic 


TROPICAL   AND   SOUTHERN  OBSERVATORIES  387 

error  varying  with  the  declination  of  the  star,  which  probably  arose  from  an 
error  in  the  collimation  of  the  transit. 

SANTIAGO,  1850. — A  Catalogue  of  1963  Stars together  with  a  Catalogue 

of  290  Double  Stars,  the  whole  from  observations  made  at  Santiago,  Chili,  dur- 
ing the  years  1850-51-52,  by  the  U.S.  Naval  Astronomical  Expedition  to  the 
Southern  Hemisphere,  LIEUT.  JAMES  M.  GILLISS,  LL.D.,  Superintendent. 
Washington  Observations  for  1868,  Appendix  I.,  Washington,  1870. 

SANTIAGO,    1855. — Catalogo    de  Ascenciones    Rectas  i  Distancias    Polares 

medias    dedueidas  de  las  observaciones  en  los  anos  1853,  1854,  i  1855. 

Santiago  de  Chile,  1859. 

SANTIAGO,  1860. — Ascenciones  Rectas  i  Distancias  Polares  de  las  estrellas 
observadas  en  los  anos  de  1856  d  1860  con  el  Circulo  Meridiano.  Observatorio 
National,  Santiago  de  Chile,  1875. 

MELBOURNE,  1870. — First  Melbourne  General  Catalogue  of  1,227  Stars  for 
the  Epoch  1870,  deduced  from  observations  extending  from  1863  to  1870,  made 
at  the  Melbourne  Observatory.  Melbourne,  1874. 

CORDOBA,  1875. — The  Argentine  General  Catalogue.  Resultados  del  Obser- 
vatorio National  Argentino,  vol.  xiv.,  Cordoba,  1886. 

MADRAS,  1875. — Results  of  Observations  of  the  Fixed  Stars  made  with  the 
Madras  Meridian  Circle,  vol.  ix.  General  Catalogue.  Madras,  1899. 

CAPE,  1840-1900. — Catalogues  constructed  from  the  observations  at  the 
Cape  have  appeared  for  the  epochs  : 

1840  From  observations       -       1834-40. 

1850  „  „  -       1849-52. 

Also,  Cape  Catalogue  q/1159  Stars,  Royal  Observatory,  Cape  of  Good  Hope, 
1856  to  1861,  reduced  to  the  epoch  1860  by  E.  J.  STONE,  1873. 

Catalogue  of  1905  Stars  for  the  Equinox  1865,  from  observations  made  at  the 
Royal  Observatory,  Cape  of  Good  Hope,  1861  to  1870,  by  SIR  THOMAS  MACLEAR. 
Reduced  by  DAVID  GILL.  8vo,  London,  1899. 

Catalogue  of  12,441  Stars  for  the  Epoch  1880,  from  observations  made  at  the 
Royal  Observatory,  Cape  of  Good  Hope,  1871-1879,  by  E.  J.  STONE.  4to, 
London,  1881. 

Catalogue  of  1713  Stars  for  the  Equinox  1885,  from  observations  made  at 
the  Royal  Observatory,  Cape  of  Good  Hope,  1879-1885.  DAVID  GILL.  4to, 
London,  1894. 

Catalogue  of  3007  Stars  for  the  Equinox  1890,  from  observations  made  at 
the  Royal  Observatory,  Cape  of  Good  Hope,  1885-1895.  DAVID  GILL.  4to, 
London,  1898. 


APPENDIX  OF  FORMULAE  AXD  TARfJSL 


7-z 


-    1 


I  CM 


390  APPENDIX  OF   FORMULAE  AND  TABLES 

The  fact  that  the  years  1600,  2000,  and  2400  are  bissextile, 
while  the  numbers  IA.  are  arranged  for  centuries  which  begin 
with  a  common  year,  makes  it  necessary  to  diminish  the  day  for 
these  years  by  1,  using  —  1  instead  of  0  for  the  year. 

Tables  II.  and  III.  will  be  readily  understood  by  a  study  of 
Chap  V.,  §§61  and  65. 

Table  IV. — Each  column  of  hundredths  of  a  day  in  this  table 
is  followed  by  a  column  containing  the  equivalent  in  hours, 
minutes,  and  seconds.  The  column  next  following  is  one 
hundredth  of  this,  and  therefore  gives  the  equivalent  for  the 
third  and  fourth  decimals  of  the  day.  The  third  column  fol- 
lowing gives  the  equivalent  for  the  fifth  and  sixth  decimals. 
As  an  example  the  reduction  of  0720  853  to  h.,  m.,  and  s.  is 

17  h.  16  m.  48  s.  + 1  m.  9'12  s.  +  4'58  s.  =  17  h.  18  m.  1'70  s. 
For  the  reverse  reduction,  we  find  in  the  second  column  of  either 
part  of  the  table  the  h.,  m.,  and  s.  next  smaller  than  the  given 
ones,  and  write  down  the  corresponding  two  figures  of  the 
argument.  Then  we  take  mentally  the  excess  of  the  given 
h.,  m.,  and  s.  above  that  of  the  table,  enter  the  third  column  for 
the  next  two  decimals,  and  so  on. 

Table  V.  gives  in  the  second  column  of  each  part  the  time  of 
beginning  of  each  solar  year  during  the  twentieth  century. 
During  the  first  72  years  of  the  century  the  moment  of 
beginning  is  always  after  Greenwich  mean  noon  of  the  zero 
date  of  the  year,  that  is  December  31  of  the  year  preceding. 
Beginning  with  the  year  1973,  the  solar  bissextile  years  begin 
before  January  0  as  thus  defined,  and  the  date  is  therefore 
negative  in  the  table.  These  data  being  especially  useful  in 
tables  and  ephemerides  of  apparent  places  of  stars,  the  other 
arguments  necessary  for  computing  these  places  are  given  in 
Tables  V.,  VI.,  and  VII.  for  sidereal  days  reckoned  from  the 
beginning  of  the  solar  year. 

As  to  the  form  of  these  tables,  it  should  be  noted  that  the 
"  tabular  year,"  frequently  used  in  astronomical  tables,  begins  on 
January  0  of  common  years  as  here,  but  on  January  1  of  leap 
years.  During  the  months  of  January  and  February  the  days 
of  this  tabular  year  are  less  by  1  than  of  the  civil  year.  But,  in 


APPENDIX  OF  FORMULAE  AND  TABLES  391 

the  present  tables,  this  tabular  year  is  not  used ;  hence  for  dates 
after  March  1  the  day  of  the  year  will  be  1  greater  in  leap  years 
than  in  common  years. 


APPENDIX  III. 

Centennial  values  of  the  precessional  motions.  —  These  motions 
are  computed  as  shown  in  Chapter  IX.,  Section  1.  The  pre- 
cessions given  in  most  catalogues  of  stars  are  annual,  and  the 
secular  variation  of  each  is  its  change  for  100  years,  and  is 
therefore  Y^-Q-  the  value  of  D'^OL  or  D^S.  It  may  be  computed 
with  the  same  coefficients,  using  the  annual  instead  of  the 
centennial  motions,  except  as  to  the  small  terms  factored  by 


If  the  secular  variation  of  the  precession  alone  is  required, 
fjia  and  /ULS  should  be  added  to  pa  and  ps  instead  of  2/u.a  and  2/xs  ; 
and  if  the  effect  of  proper  motions  is  to  be  entirely  omitted, 
as  in  reducing  the  geometric  place  unchanged,  jua  and  ^  should 
be  taken  as  zero. 


APPENDIX  IV. 

The  development  of  the  methods  set  forth  in  this  appendix  is 
fully  given  in  Chapter  X.,  Section  II.,  where  examples  of  the 
use  of  the  tables  will  be  found. 


APPENDIX   V. 

In  most  star-catalogues  between  1850  and  1900  the  pre- 
cessions are  those  of  Struve-Peters.  They  may  be  reduced  to 
the  new  values  by  applying  —38  to  the  number  from  the  first 
column  of  Table  XVI.  to  obtain  Apa,  and  multiplying  the  number 
from  the  second  column  of  XVI.  by  nat.  tan  6  from  XVII.,  which 
will  give  A/Ja-  If  the  precessions  are  annual,  the  units  of  the 
correction  will  then  be  Os'00001  and  (T0001,  respectively;  if 
centennial,  Os'001  and  0"'01. 


392  APPENDIX   OF   FORMULAE   AND  TABLES 

APPENDIX  VI. 

These  tables  are  used  for  the  rapid  conversion  of  ecliptic  into 
equatorial  coordinates  and  vice  versa,  where  a  greater  accuracy 
than  to  a  coarse  fraction  of  a  minute  is  not  required. 

Table  XX.  is  arranged  for  the  rapid  conversion  of  small  correc- 
tions in  one  set  of  coordinates  to  corrections  in  the  other.  As 
this  conversion  is  rarely  necessary,  except  in  the  case  of  the 
moon  and  planets,  it  is  given  only  between  the  limits  —  5°  and 
+  5°  of  latitude. 

APPENDIX  VII. 

The  condensed  table  of  refraction  here  given  is  only  approxi- 
mate. Refractions  correct  to  +  0"'l  may,  however,  be  found 
from  it  when  the  zenith  distance  is  not  too  great,  and  the 
deviation  of  the  temperature  and  pressure  from  the  adopted 
standard  not  too  wide. 


APPENDIX  IX. 

Three-place  tables  of  logarithms  and  trigonometric  functions 
are  given,  because  they  are  not  usually  at  hand,  and  should  be 
used  in  all  cases  when  sufficiently  accurate.  It  is  often  easier 
to  form  a  product  of  three  figures  by  three  with  numbers  than 
by  logarithms  especially  if  a  table  of  products  is  used.  The 
natural  values  of  the  trigonometric  are  therefore  often  con- 
venient to  use  instead  of  their  logarithms.  But  the  latter  are. 
preferable  in  forming  a  product  of  more  than  two  factors. 


APPENDIX   I. 


CONSTANTS  AND  FORMULAE  IN  FREQUENT  USE. 


Ratio  of 

Degrees 
Minutes 

A.—  Constants  with  their  Logarithms. 

Numbers.             Logarithm; 
circumference  to  diameter  TT  =  3*141  592  65       0-497  149 
2;r  =  6-283  185  31       0*798179 
7T2-  9  -869  60440       0-994299 
JTT  =  1-772  453  85       0*248  574 

in  circumference                                   360            2-556  302 
21  600            4-334  453 

3. 

9 
9 

7 
9 

5 

8 

Seconds 

5) 

-      1  296  000 

6 

•112 

605 

0 

Degrees 

in  radian   - 

-  57°-295  779  5 

1 

•758 

122 

6 

Minutes 

M 

3  437'  -746  77 

3 

•536 

273 

9 

Seconds 

)j 

-  206  264"-806 

5 

•314 

425 

1 

Seconds 

of  time  in  radian 

13  750s-987 

4 

•138 

333 

9 

Length  of  arc  of  one  degree 

-     0-017  453  29 

8 

•241 

877 

4- 

10 

55 

„     minute 

0-000  290  89 

6 

•463 

726 

1- 

10 

5) 

„     second 

0-000  004  848 

4 

•685 

574 

9- 

10 

)J 

„          „     second 

of  time      0-000  072  722 

5 

•861 

666 

1  — 

10 

Hours  in  one  day 

24 

1 

•380 

211 

2 

Minutes 

5) 

1440 

3 

•158 

362 

5 

Seconds 

)) 

86400 

4 

•936 

513 

7 

Days  in 

Julian  Year 

365-25 

2 

•562 

590 

2 

Hours 

n 

8766 

3 

•942 

801 

5 

Minutes 

?) 

525  960 

5 

•720  952 

7 

Seconds 

» 

31  557  600 

7 

•499 

104 

0 

Days  in 

Solar  Year 

365-2422 

2 

•562 

580 

9 

Hours 

j» 

8  765-813 

3 

•942 

792 

2 

Minutes 

55 

525  948-77 

5 

•720 

943 

4 

Seconds 

>J 

31  556  926-0 

7 

•499  094 

7 

The  following  values  of  the  constants  for  reduction  of  places  of  the 
fixed  stars  were  most  in  use  between  1830  and  1900,  but  are  now  being 
superseded  by  the  values  adopted  in  the  present  work. 


394  APPENDIX   I. 

The  values  of  the  variable  quantities  are  given  for  1850,  from  which 
epoch  T  is  reckoned  in  centuries. 

Annual  general  precession,  Bessel  -  50"-2357  +  '0244  T 

„  „  „       Peters-Struve  50  -2524  +  -0227  T 

„       motion  of  pole,  Bessel  20  -0547  -  '0097  T 

.„  „  „     Peters-Struve  20  -0564  -  -0086  T 

Constant  of  nutation,  Bessel  -  8"-977 

„                   „           Peters-Struve  9  -223 

Constant  of  aberration,  Bessel  20"-255 

„                   „              Struve  20  -445 

Nyren  20-492 

Newcomb  -  20  -50 

Chandler  20  -53 

Dimensions   of    the    geoid    according    to   the   leading    authorities. 
Helmert's  b  here  given  is  his  latest  result.    ^ean  Ra(jiUs 

a,  metres.  b,  metres.  Metres.  Compression. 

Helmert         6378000         6356612         6370843         1  -f  298-20 
Clarke  6378249         6356515         6370997         1 -s- 293-46 

Bessel  6377397         6356079         6370282         1 4- 299-15 

B.— Formulae  for  the  Solution  of  Spherical  Triangles. 

a,      b,      c     the  sides. 
A,     B,     C    the  opposite  angles. 

CASE  I. — Given  two  sides  a,  b  and  the  included  angle  C. 
sin  c  sin  A  —  sin  a  sin  C, 
sin  c  cos  A  =  cos  a  sin  b  -  sin  a  cos  b  cos  (7, 
cos  c  =  cos  a  cos  b  +  sin  a  sin  b  cos  C. 
If  we  compute  k  and  K  from 

k  sin  K=  sin  a  cos  (7, 
kcosK=cosa, 

then  sin  c  cos  A  =  k  sin  (b  -  K), 

cos  c  =  k  cos  (b  -  K). 

The  Gaussian  equations  for  this  case,  not  advantageous  unless  A,  B, 
and  c  are  all  required,  are 

sin  \c  sin  \(A  -B)  =  cos  \G  sin  |(a  -  b), 
sin  Jc  cos  ^(A  -  B}  =  sin  \C  sin  J(«  +  b), 
cos  \c  sin  \(A  +B)  =  cos  |(7cos  J(a  -  6), 
cos  Jc  cos  \(A  +!>)  =  sin  J(7  cos  J(a  +  b). 


APPENDIX   I.  395 

CASE  II.  —  Given  two  angles  and  the  intermediate  side  A,  B,  c. 
sin  C  sin  a  =  sin  A  sin  c, 
sin  C  cos  a  =  cos  A  sin  B  +  sin  A  cos  B  cos  c, 

cos  (7  =  -  cos  ^  cos  B  +  sin  ^4  sin  B  cos  c. 
If  we  compute  h  and  #  from 

h  sin  Zf=  cos  A, 
h  cos  J?  =  sin  A  cos  c, 
then  sin  Ccosa  =  hcos(B  -  H), 


The  Gaussian  equations  for  this  case  are  formed  by  writing  those  of 
Case  I.  in  the  order  2,  4,  1,  3,  interchanging  the  two  members  of  each 
equation. 

CASE  III.  —  Given  the  three  sides. 


tan    A  =  —. 


2  _  sin  (s  -  a)  sin  (s  -  b)  sin  (s  -  c) 

sins 
m 


—.  —     —  r, 
sin  (s  -  a) 

-i   r>  ™ 

tanj£=   ,—j  -  TV, 

sin  (s  -  b) 

tan  1(7=-.    f     x. 
sin  (s  —  c) 

CASE  IV.  —  Given  the  three  angles. 


..^  ~     COS    S 

=  cos(S-A)  cos  (S  -  B)  cos  (S  -  C)' 
tan  \a  —  Mcos  (S  —  A), 


CASE  V.  —  Given  two  sides  and  the  angle  opposite  one  of  them, 
a,  b,  A. 

.     D     sin  A  sin  b  ,          .  m 

sin  B  =  --  :  —  (two  values  of  B  ), 


Bn  a 


,  ^    cos  A(a  —  b)     ,  ,  ,  4      -rv. 
tan  \C=  -  ^7  -  A  cot  MA  +  B\ 
cos  J(a  +  6) 


cos 


396  APPENDIX    I. 

CASE  VI.  —  Given  two  angles  and  the  side  opposite  one  of  them, 
A,  B,  a. 

.    ,     sin  a  sin  B 

sin  b  =-     ,  —  -.  —     (two  values  of  b), 
sin  A 


cos 

tan«fl 


In  right  spherical  triangles  the  fundamental  equations  take  the 
following  forms  : 

c  the  hypothenuse. 

sin  c  sin  A  =  sin  a, 
sin  c  cos  A  =  cos  a  sin  b, 

cos  c  =  cos  a  cos  #, 

sin  &  =  sin  B  sin  c, 

sin  a  cos  5  =  cos  B  sin  c, 

tan  a  =  cos  ^  tan  c, 

tan  a  =  tan  A  sin  ft. 

0.  —  Differentials  of  the  Parts  of  a  Spherical  Triangle. 

The  partial  derivatives  of  any  part  of  a  spherical  triangle  with 
respect  to  the  three  other  parts  on  which  it  depends  are  found  from 
that  one  of  the  following  equations  which  contains  the  differentials  of 
the  four  variable  parts.  (Comp.  §  6.) 

-  sin  Cda  +  cos  b  sin  Adc  +  sin  bdA  +  cos  C  sin  adB  =  0, 

-  sin  Adb  +  cos  c  sin  Bda  +  sin  cdB  +  cos  A  sin  bdC  =  Q, 

-  sin  Bdc  +  cos  a  sin  Cdb  +  sin  adC  +  cos  B  sin  oLrf  =  0. 

-  sin  b  sin  Cda  +  dA  +  cos  cd.Z?  +  cos  bdC  =  0, 

-  sin  c  sin  ^d&  +  dB  +  cos  odC  +  cos  cdA  =  0, 

-  sin  a  sin  IWe  +  dC  +  cos  6cL4  +  cos  adB  =  0. 

-  da  +  cos  CS6  +  cos  Bdc  +  sin  c  sin  BdA  =  0, 
-db  +  cos  ^c?c  +  cos  C^a  +  sin  a  sin  CdB  =  0, 
-dc  +  cos  $da  +  cos  Adb  +  sin  6  sin  ^c£(7  =  0. 

The  remaining  forms  may  be  written  by  leaving  any  one  pair  of 
letters,  say  a  and  A,  unaltered  and  interchanging  the  other  two,  say 
B  with  C  and  b  with  c. 


APPENDIX  II. 

To  find  the  Day  of  the  Julian  Period  corresponding  to  any  Day  of  the  Julian 
Calendar  to  1600,  or  of  the  Gregorian  Calendar  after  1600. 


TABLE  IA. 
FOR  CENTURY. 

TABLE  IB. 
FOR  YEAR  IN  CENTURY. 

TABLE  Ic. 

FOR  THE  DAY. 

-1900 

1  027  083 

00 

0  or  -  1 

50 

18262 

Common 

Leap 

-1800 

1  063  608 

01 

365 

51 

18627 

Year. 

Year. 

-1700 

1  100  133 

02 

730 

52 

18992 

Jan.  0 

-1600 

1  136  658 

03 

1095 

53 

19  358 

10 

10 

10 

-1500 

1  173  183 

04 

1460 

54 

19723 

20 
30 

20 

30 

20 
30 

-1400 

1  209  708 

05 

1826 

55 

20088 

Ol 

-1300 

1  246  233 

06 

2191 

56 

20  453 

Feb.  0 

31 

31 

-1200 

I  282  758 

07 

2556 

57 

20819 

10 

41 

41 

-1100 

1  319  283 

08 

2921 

58 

21  184 

20 

51 

51 

-1000 

1  355  808 

09 

3287 

59 

21549 

Mar.  0 

59 

60 

-900 

1  392  333 

10 

3652 

60 

21914 

10 

69 

70 

-800 

1  428  858 

11 

4017 

61 

22280 

20 

79 

80 

f\f\ 

-700 

1  465  383 

12 

4382 

62 

22645 

30 

89 

\)(J 

-600 

1  501  908 

13 

4748 

63 

23010 

Apr.  0 

90 

91 

-500 

1  538  433 

14 

5113 

64 

23  375 

10 

100 

101 

-400 

1  574  958 

15 

5478 

65 

23741 

20 

110 

111 

-300 

1611483 

16 

5843 

66 

24106 

May  0 

120 

121 

-200 

1  648  008 

17 

6209 

67 

24471 

10 

130 

131 

-100 

1  684  533 

18 

6574 

68 

24836 

20 

140 

141 

0 

1  721  058 

19 

6939 

69 

25202 

30 

150 

151 

+  100 

1  757  583 

20 

7304 

70 

25567 

June  0 

151 

152 

200 

1  794  108 

21 

7670 

71 

25932 

10 

161 

162 

300 

1  830  633 

22 

8035 

72 

26297 

20 

171 

172 

400 
500 

1  867  158 
1  903  683 

23 
24 

8400 
8765 

73 
74 

26663 

27028 

July  0 
10 

181 
191 

182 
192 

600 

1  940  208 

25 

9131 

75 

27393 

20 

201 

202 

700 

1  976  733 

26 

9496 

76 

27758 

30 

211 

212 

800 
900 
1000 

2  013  258 
2  049  783 
2  086  308 

27 
28 
29 

9861 
10  226 
10592 

77 
78 
79 

28124 

28489 

28854 

Aug.  0 
10 
20 

212 
222 
232 

213 
223 
233 

1100 

2  122  833 

30 

10957 

80 

29219 

30 

242 

243 

1200 
1300 

2  159  358 
2  195  883 

31 
32 

11322 
11687 

81 
82 

29585 
29950 

Sept.  0 

243 

244 

1400 
1500 

2  232  408 
2  268  933 

33 
34 

12  053 
12418 

83 
84 

30315 

30680 

10 
20 

253 
263 

254 
264 

1600 

2  305  448 

35 

12  783 

85 

31046 

Oct.  0 

273 

274 

1700 
1800 
1900 

2  341  972 

2  378  496 
2415020 

36 
37 
38 

13148 
13  514 
13879 

86 
87 
88 

31411 
31  776 
32  141 

10 
20 
30 

283 
293 
303 

284 
294 
304 

2000 

2451545 

39 

14244 

89 

32507 

NOV.  0 

304 

305 

2100 
2200 

2  488  069 
2  524  593 

40 
41 

14609 
14975 

90 
91 

32  872 
33237 

10 
20 

314 
324 

315 
325 

2300 

2561117 

42 

15  340 

92 

33  602 

Dec.  o 

334 

335 

2400 

2  597  642 

43 

15705 

93 

33  968 

10 

344 

345 

44 

16070 

94 

34333 

20 

354 

355 

45 

16436 

95 

34698 

30 

364 

365 

46 

16801 

96 

35063 

47 

17166 

97 

35429 

48 

17  531 

98 

35794 

49 

17897 

99 

36159 

398 


APPENDIX   II. 


TABLE   HA. 

To  Convert  Mean  into  Sidereal  Time. 


Mean 
Time. 

Correction. 

Mean 
Time. 

Correction. 

Mean 
Time. 

Correction. 

Correction 
for  Minutes  and 
Seconds. 

h.      m. 

m.          s. 

h.      m. 

m.        s. 

h.       m. 

m.        s. 

m.     s. 

s. 

0     o 

o    o-oo 

8     o 

1     18-85 

16     o 

2    37-70 

0     10 

0-03 

10 

1-64 

10 

20-50 

10 

39-35 

20 

0'05 

20 

3-29 

20 

22-14 

20 

40-99 

30 

0'08 

30 

4-93 

30 

23-78 

30 

42-63 

40 
50 

O']l 
0'14 

40 

6-57 

40 

25-42 

40 

44-28 

50 

8-21 

50 

27-07 

50 

45-92 

1       0 

0-16 

10 

0'19 

1     o 

0      9-86 

9     o 

1     28-71 

17      o 

2    47-56 

20 

0-22 

10 

11-50 

10 

30-35 

10 

49-20 

30 

0-25 

20 

13-14 

20 

31-99 

20 

50-85 

40 

0-27 

30 

14-78 

30 

33-64 

30 

52-49 

50 

0-30 

40 

16-43 

40 

35-28 

40 

54-13 

2       0 

0-33 

50 

18-07 

50 

36-92 

50 

55-77 

10 

0-36 

2     o 

0    19-71 

10     o 

1     38-56 

18     o 

2    57-42 

20 
30 

0-38 
0'41 

10 

21-36 

10 

40-21 

10 

5906 

40 

0-44 

20 

23-00 

20 

41-85 

20 

3      0-70 

50 

0-47 

30 

24-64 

30 

43-49 

30 

2-34 

40 

26"28 

40 

45-14 

40 

3-99 

3       0 
in 

0*49 

O'SO 

50 

27-93 

50 

46-78 

50 

5-63 

jyu 
20 

o^ 
0-55 

3     o 

0    29-57 

11     o 

1     48-42 

19     o 

3      7-27 

30 

0-57 

10 

31-21 

10 

50-06 

10 

8-92 

40 

0'60 

20 

32-86 

20 

51-71 

20 

10-56 

50 

0'6o 

30 

34-50 

30 

53-35 

30 

12-20 

4       0 

0-66 

40 

36-14 

40 

54-99 

40 

13-84 

10 

0-68 

50 

37-78 

50 

56-64 

50 

15-49 

20 

0-71 

30 

0'74 

4      o 

0    39-43 

12       0 

1     58-28 

20     o 

3     17-13 

40 

0-77 

10 

41-07 

10 

59-92 

10 

18-77 

50 

079 

20 

42-71 

20 

2       1-56 

20 

20-42 

K       0 

0*82 

30 

44-35 

30 

3-21 

30 

22-06 

tl             v 

10 

0'85 

40 

46-00 

40 

4-85 

40 

23-70 

20 

0-88 

50 

47-64 

50 

6-49 

50 

25-34 

30 

0-90 

5     o 

0    49-28 

13      o 

2       8-13 

21      o 

3    26-99 

40 
50 

0-93 
0'96 

10 

50-92 

10 

9-78 

10 

28-63 

20 

52-57 

20 

11-42 

20 

30-27 

6      0 

0-99 

30 

54-21 

30            13-06 

30 

31-91 

10 

1-01 

40 

55-85 

40 

14-70 

40 

33-56 

20 

1'04 
1  '07 

50 

57-50 

50 

16-35 

50 

35-20 

40 

J.  \Ji 
1-10 

6     o 

0    59-14 

14     o 

2     17-99 

22      o 

3    36-84 

50 

1-12 

10 

1      0-78 

10  1         19-63 

10 

38-48 

7       0 

1-15 

20 

2-42 

20  ;         21-28 

20 

40-13 

10 

1-18 

30 

4-07 

30  i         22  92 

30 

41-77 

20 

1-21 

40 

5-71 

40  i         24-56 

40 

43-41 

30 

1-23 

50 

7-35 

50 

26-20 

50 

45-06 

40 

1-26 

50 

1-29 

7     o 

1       9-00 

15     o 

2    27-85 

23     o 

3    46-70 

1»O1 

10 

10-64 

10 

29-49 

10 

48-34 

8       0 

ol 

i  ••  •  i 

20 

12-28 

20 

31-13 

20 

49-98 

10 

9O 

1  o4 

30 

13-92 

30 

32-77 

30 

51-63 

Up 

30 

1'40 

40 

15-57 

40  !         34-42 

40 

53-27 

40 

1-42 

50 

17-21 

50           36-06 

50 

54-91 

50 
9n 

1-45 

0 
10 

1-50 

20 

1-53 

30 

1-56 

40 

1-59 

50 

1-62 

APPENDIX   II. 


399 


TABLE    IlB. 

To  Convert  Sidereal  into  Mean  Time. 


Sidereal 
Time. 

Correction. 

Sidereal 
Time. 

Correction. 

Sidereal 
Time. 

Correction. 

Correction 
for  Minutes  and 
Seconds. 

h.      m. 

m.          s. 

h.      m. 

m.         s. 

h.        m. 

rn.        s. 

m.     s. 

s. 

0     o 

o    o-oo 

8     o 

1     18-64 

16      o 

2    37-27 

0     10 

0-03 

10 

1-64 

10 

20-28 

10 

38-91 

20 

0-05 

20 

3-28 

20 

21-91 

20 

40-55 

30 

0'08 

30 

4-92 

30 

23-55 

30 

42-19 

40 

•A 

Oil 

0.-IA 

40 

6-55 

40 

25-19 

40 

43-83 

DU 

14 

50 

8-19 

50 

26-83 

50 

45-46 

1       0 

0-16 

10 

0'19 

1       0 

0      9-83 

9     o 

1     28-47 

17      o 

2    47-10 

20 

0-22 

10 

11-47 

10 

30-10 

10 

48-74 

30 

0-25 

20 

13-11 

20 

31-74 

20 

50-38 

40 

0-27 

30 

14-74 

30 

33-38 

30 

52-02 

50 

0-30 

40 

16-38 

40 

3502 

40 

53-66 

2       0 

0-33 

50 

18-02 

50 

36-66 

50 

55-29 

10 

0-35 

2       0 

0     19-66 

10     o 

1     38-30 

18     o 

2    56-93 

20 

on 

0-38 

(V4.1 

10 

21-30 

10 

39-93 

10 

58-57 

OU 

40 

U  •*! 

0-44 

20 

22-94 

20 

41-57 

20 

3      0-21 

50 

0-47 

30 

24-57 

30 

43-21 

30 

1-85 

40 

26-21 

40 

44-85 

40 

3-48 

3       0 

0-49 

50 

27-85 

50 

46-49 

50 

5-12 

10 
20 

O'o2 
0-55 

3     o 

0    29-49 

11     o 

1     48-12 

19     o 

3      6-76 

30 

0-57 

10 

31-13 

10 

49-76 

10 

8-40 

40 

0-60 

20 

32-76 

20 

51-40 

20 

10-04 

50 

0'63 

30 

34-40 

30 

53-04 

30 

11-68 

4       0 

0-66 

40 

36-04 

40 

54-68 

40 

13-32 

10 

0-68 

50 

37-68 

50 

56-32 

50 

14-95 

20 

0-71 

30 

0-74 

4       0 

0    39-32 

12       0 

1     57-96 

20     o 

3     16-59 

40 

0-76 

10 

40-96 

10 

59-59 

10 

18-23 

50 

0-79 

20 

42-60 

20 

2       1-23 

20 

19-87 

50 

0'82 

30 

44-23 

30 

2-87 

30 

21-51 

V 

10 

0'85 

40 

45-87 

40 

4-51 

40 

23-14 

20 

0-87 

50 

47-51 

50 

6-15 

50 

24-78 

30 

0-90 

5       0 
10 

0    49-15 
50-79 

13      o 

10 

2      7-78 
9-42 

21      o 

10 

3     26-42 
28-06 

40 
50 

0-93 
0-96 

20 

52-42 

20 

11-06 

20 

29-70 

6       0 

098 

30 

54-06 

30 

12-70 

30 

31-34 

10 

1-01 

40 
50 

55-70 
57-34 

40 
50 

14-34 
15-98 

40 
50 

32-97 
34-61 

20 
30 
40 

1'04 
1-06 
1-09 

6       0 

0    58-98 

14       0 

2    17-61 

22      o 

3     36-25 

50 

112 

10 

1       0-62 

10 

19-25 

10 

37-89 

7       0 

115 

20 

2-25 

20 

20-89 

20 

39-53 

10 

117 

30 

3-89 

30 

22-53 

30 

41-16 

20 

1-20 

40 

5-53 

40 

24-17 

40 

42-80 

30 

1-23 

50 

7-17 

50 

25-80 

50 

44-44 

40 

1-26 

50 

1'28 

7     o 

1      8-81 

15     o 

2    27-44 

23      o 

3    46-08 

10 

10-44 

10 

29-08 

10 

47-72 

8       0 

1-31 

20 

12-08 

20 

30-72 

20 

49-36 

10 

OA 

1-34 

1   .'•>  — 

30 

13-72 

30 

32-36 

30 

51-00 

ml 

30 

1  37 
1*39 

40 

15-36 

40 

34-00 

40 

52-63 

40 

1-42 

50 

17-00 

50 

35-64 

50 

54-27 

50 

1-45 

0 
10 

1-47 
1-50 

20 

1-53 

30 

1-56 

40 

1-58 

50 

1-61 

400 


APPENDIX   II 


TABLE  III. 

Time  into  Arc  and  vice  versa. 


h.  m. 

0 

h.  m. 

0 

h.  m. 

0 

h.  m. 

0 

h.  m. 

0 

h.  m. 

0 

0  0 

0 

3  0 

45 

6  0 

90 

9  0 

135 

12  0 

180 

15  0 

225 

4 

1 

4 

46 

4 

91 

4 

136 

4 

181 

4 

226 

8 

2 

8 

47 

8 

92 

8 

137 

8 

182 

8 

227 

12 

3 

12 

48 

12 

93 

12 

138 

12   183 

12 

228 

16 

4 

16 

49 

16 

94 

16 

139 

16   184 

16 

229 

0  20 

5 

3  20 

50 

6  20 

95 

9  20 

140 

12  20   185 

15  20 

230 

24 

6 

24 

51 

24 

96 

24 

141 

24 

186 

24 

231 

28 

7 

28 

52 

28 

97 

28 

142 

28 

187 

28 

232 

32 

8 

32 

53 

32 

98 

32 

143 

32 

188 

32 

233 

36 

9 

36 

54 

36 

99 

36 

144 

36 

189 

36 

234 

0  40 

10 

3  40 

55 

6  40 

100 

9  40 

145 

12  40 

190 

15  40 

235 

44 

11 

44 

56 

44 

101 

44 

146 

44 

191 

44 

236 

48 

12 

48 

57 

48 

102 

48 

147 

48 

192 

48 

237 

52 

13 

52 

58 

52 

103 

52 

148 

52 

193 

52 

238 

56 

14 

56 

59 

56 

104 

56 

149 

56 

194 

56 

239 

1  0 

15 

4  0 

60 

7  0 

105 

10  0 

150 

13  0 

195 

16  0 

240 

4 

16 

4 

61 

4 

106 

4 

151 

4 

196 

4 

241 

8 

17 

8 

62 

8 

107 

8 

152 

8 

197 

8 

242 

12 

18 

12 

63 

12 

108 

12 

153 

12 

198 

12 

243 

16 

19 

16 

64 

16 

109 

16 

154 

16 

199 

16 

244 

1  20 

20 

4  20 

65 

7  20 

110 

1020 

155 

13  20 

200 

16  20  245 

24 

21 

24 

66 

24 

111 

24 

156 

24 

201 

24 

246 

28 

22 

28 

67 

28 

112 

28 

157 

28 

202 

28 

247 

32 

23 

32 

68 

32 

113 

32 

158 

32 

203 

32 

248 

36 

24 

36 

69 

36 

114 

36 

159 

36 

204 

36 

249 

1  40 

25 

4  40 

70 

7  40 

115 

10  40 

160 

13  40 

205 

16  40 

250 

44 

26 

44 

71 

44 

116 

44 

161 

44 

206 

44 

251 

48 

27 

48 

72 

48 

117 

48 

162 

48  207 

48 

252 

52 

28 

52 

73 

52 

118 

52 

163 

52  208 

52 

253 

56 

29 

56 

74 

56 

119 

56 

164 

56 

209 

56 

254 

2  0 

30 

5  0 

75 

8  0 

120 

11  0 

165 

14  0 

210 

17  0 

255 

4 

31 

4 

76 

4 

121 

4 

166 

4  211 

4 

256 

8 

32 

8 

77 

8 

122 

8 

167 

8  1  212 

8 

257 

12 

33 

12 

78 

12 

123 

12 

168 

12 

213 

12 

258 

16 

34 

16 

79 

16 

124 

16 

169 

16 

214 

16 

259 

2  20 

35 

5  20 

80 

8  20 

125 

11  20 

170 

14  20 

215 

17  20 

260 

24 

36 

24 

81 

24 

126 

24 

171 

24 

216 

24 

261 

28 

37 

28 

82 

28 

127 

28 

172 

28 

217 

28 

262 

32 

38 

32 

83 

32 

128 

32 

173 

32 

218 

32 

263 

36 

39 

36 

84 

36 

129 

36 

174 

36 

219 

36 

264 

2  40 

40 

5  40 

85 

8  40 

130 

11  40 

175 

14  40 

220 

17  40 

265 

44 

41 

44 

86 

44 

131 

44 

176 

44 

221 

44 

266 

48 

42 

48 

87 

48 

132 

48 

177 

48 

222 

48 

267 

52 

43 

52 

88 

52 

133 

52 

178 

52 

223 

52 

268 

56 

44 

56 

89 

56 

134 

56 

179 

56 

224 

56 

269 

3  0 

46 

6  0 

90 

9  0 

135 

12  0 

180 

15  0 

225 

18  0 

270 

APPENDIX   II. 


401 


TABLE  III— Concluded. 
Time  into  Arc  and  vice  versa. 


h.  m. 

0 

h.  rn. 

0 

m.  s. 

/ 

m.  s. 

/ 

'S. 

„ 

s. 

// 

18  0 

270 

21  0 

315 

o  o 

0 

2  0 

30 

o-ooo 

0 

2-000 

30 

4 

271 

4 

316 

4 

1 

4 

31 

0-066 

1 

2-066 

31 

8 

272 

8 

317 

8 

2 

•   8 

32 

0-133 

2 

2-133 

32 

12 

273 

12 

318 

12 

3 

12 

33 

0-200 

3 

2-200 

33 

16 

274 

16 

319 

16 

4 

16 

34 

0-266 

4 

2-266 

34 

18  -20 

275 

21  20 

320 

0  20 

5 

2  20 

35 

0-333 

5 

2-333 

35 

24 

276 

24 

321 

24 

6 

24 

36 

0-400 

6 

2-400 

36 

28 

277 

28 

322 

28 

7 

28 

37 

0-466 

7 

2-466 

37 

32 

278 

32 

323 

32 

8 

32 

38 

0-533 

8 

2-533 

38 

36 

279 

36 

324 

36 

9 

36 

39 

0-600 

9 

2-600 

39 

18  40 

280 

21  40 

325 

0  40 

10 

2  40 

40 

0-666 

10 

2-666 

40 

44 

281 

44 

326 

44 

11 

44 

41 

0-733 

11 

2-733 

41 

48 

282 

48 

327 

48 

12 

48 

42 

0-800 

12 

2-800 

42 

52 

283 

52 

328 

52 

13 

52 

43 

0-866 

13 

2-866 

43 

56 

284 

56 

329 

56 

14 

56 

44 

0-933 

14 

2-933 

44 

19  0 

285 

22  0 

330 

1  0 

15 

3  0 

45 

1-000 

15 

3-000 

45 

4 

286 

4 

331 

4 

16 

4 

46 

1-066 

16 

3-066 

46 

8 

287 

8 

332 

8 

17 

8 

47 

1-133 

17 

3-133 

47 

12 

288 

12 

333 

12 

18 

12 

48 

1-200 

18 

3-200 

48 

16 

289 

16 

334 

16 

19 

16 

49 

1-266 

19 

3-266 

49 

19  20 

290 

22  20 

335 

1  20 

20 

3  20 

50 

1-333 

20 

3-333 

50 

24 

291 

24 

336 

24 

21 

24 

51 

1-400 

21 

3-400 

51 

28 

292 

28 

337 

28 

22 

28 

52 

1-466 

22 

3-466 

52 

32 

293 

32 

338 

32 

23 

32 

53 

1  -533 

23 

3-533 

53 

36 

294 

36 

339 

36 

24 

36 

54 

1  -600 

24 

3-600 

54 

19  40 

295 

22  40 

340 

1  40 

25 

3  40 

55 

1-666 

25 

3-666 

55 

44 

296 

44 

341 

44 

26 

44 

56 

1-733 

26 

3-733 

56 

48 

297 

48 

342 

48 

27 

48 

57 

1-800 

27 

3-800 

57 

52 

298 

52 

343 

52 

28 

52 

58 

1-866 

28 

3-866 

58 

56 

299 

56 

344 

56 

29 

56 

59 

1-933 

29 

3-933 

59 

20  0 

300 

23  0 

345 

2  0 

30 

4  0 

60 

2-000 

30 

4-000 

60 

4 

301 

4 

346 

8 

302 

8 

347 

12 

303 

12 

348 

16 

304 

16 

349 

20  20 

305 

23  20 

350 

24 

306 

24 

351 

28 

307 

28 

352 

32 

308 

32 

353 

36 

309 

36 

354 

20  40 

310 

23  40 

355 

44 

311 

44 

356 

48 

312 

48 

357 

52 

313 

52 

358 

56 

314 

56 

359 

21  0 

315 

24  0 

360 

N.S.A. 


402 


APPENDIX   II. 


TABLE   IV. 

To  change  Decimals  of  a  Day  to  Hours,  Minutes,  and  Seconds,  and 


vice  versa. 


d. 

h.  m.  s. 

m.   s. 

s. 

d. 

h.  m.  s. 

m.   s. 

s. 

o-oi 

0  14  24 

0  8-64 

0-09 

0-51 

12  14  24 

7  20-64 

4-41 

0-02 

0  28  48 

0  17-28 

0-17 

0-52 

12  28  48 

7  29-28 

4-49 

0-03 

0  43  12 

0  25-92 

0-26 

0-53 

12  43  12 

7  37-92 

4-58 

0-04 

0  57  36 

0  34-56 

0-35 

0-54 

12  57  36 

7  46-56 

4-67 

0-05 

1  12  0 

0  43-20 

0-43 

0-55 

13  12   0 

7  55-20 

4-75 

0-06 

1  26  24 

0  51-84 

0-52 

0-56 

13  26  24 

8  3-84 

4-84 

0-07 

1  40  48 

0-48 

0-60 

0-57 

13  40  48 

8  12-48 

4-92 

0-08 

1  55  12 

9-12 

0-69 

0-58 

13  55  12 

8  21-12 

5-01 

0-09 

2  9  36 

17-76 

0-78 

0-59 

14  9  36 

8  29-76 

5-10 

o-io 

2  24  0 

26-40 

0-86 

0-60 

14  24  0 

8  38-40 

5-18 

O'll 

2  38  24 

35-04 

0-95 

061 

14  38  24 

8  47-04 

5-27 

0-12 

2  52  48 

43-68 

1-04 

0-62 

14  52  48 

8  55-68 

5-36 

0-13 

3  7  12 

1  52-32 

1-12 

0-63 

15   7  12 

9  4-32 

5-44 

0-14 

3  21  36 

2  0-96 

1-21 

0-64 

15  21  36 

9  12-96 

5-53 

0-15 

3  36  0 

2  9-60 

1-30 

0-65 

15  36   0 

9  21-60 

5-62 

0-16 

3  50  24 

2  18-24 

1-38 

0-66 

15  50  24 

9  30-24 

5-70 

0-17 

4  4  48 

2  26-88 

1-47 

0-67 

16  4  48 

9  38-88 

5-79 

0-18 

4  19  12 

2  35-52 

1-56 

0-68 

16  19  12 

9  47-52 

5-88 

0-19 

4  33  36 

2  44-16 

1-64 

0-69 

16  33  36 

9  56-16 

5-96 

0-20 

4  48  0 

2  52-80 

1-73 

0-70 

16  48  0 

10  4-80 

6-05 

0-21 

5   2  24 

3  1-44 

1-81 

071 

17   2  24 

10  13-44 

6-13 

0-22 

5  16  48 

3  10-08 

1-90 

0-72 

17  16  48 

10  22-08 

6-22 

0-23 

5  31  12 

3  18-72 

1-99 

0-73 

17  31  12 

10  30-72 

6-31 

0-24 

5  45  36 

3  27-36 

2-07 

0-74 

17  45  36 

10  39-36 

6-39 

0-25 

600 

3  36-00 

2-16 

0-75 

18   0  0 

10  48-00 

6-48 

0-26 

6  14  24 

3  44-64 

2-25 

0-76 

18  14  24 

10  56-64 

6-57 

0-27 

6  28  48 

3  53-28 

2-33 

0-77 

18  28  48 

11  5-28 

6-65 

0-28 

6  43  12 

4  1-92 

2-42 

0-78 

18  43  12 

11  13-92 

6-74 

0-29 

6  57  36 

4  10-56 

2-51 

0-79 

18  57  36 

11  22-56 

6-83 

0-30 

7  12   0 

4  19-20 

2-59 

0-80 

19  12   0 

11  31-20 

6-91 

0-31 

7  26  24 

4  27-84 

2-68 

0-81 

19  26  24 

11  39-84 

7-00 

0-32 

7  40  48 

4  36-48 

2-76 

0-82 

19  40  48 

11  48-48 

7-08 

0-33 

7  55  12 

4  45-12 

2-85 

0-83 

19  55  12 

11  57-12 

7-17 

0-34 

8  9  36 

4  53-76 

2-94 

0-84 

20  9  36 

12  5-76 

7-26 

0-35 

8  24  0 

5  2-40 

3-02 

0-85 

20  24  0 

12  14-40 

7-34 

0-36 

8  38  24 

5  11-04 

3-11 

0-86 

20  38  24 

12  23-04 

7-43 

0-37 

8  52  48 

5  19-68 

3-20 

0-87 

20  52  48 

12  31-68 

7-52 

0-38 

9   7  12 

5  28-32 

3-28 

0-88 

21   7  12 

12  40-32 

7-60 

0-39 

9  21  36 

5  36-96 

3-37 

0-89 

21  21  36 

12  48-96 

7-69 

0-40 

9  36  0 

5  45-60 

3-46 

0-90 

21  36  0 

12  57-60 

7-78 

0-41 

9  50  24 

5  54-24 

3-54 

0-91 

21  50  24 

13  6-24 

7-86 

0-42 

10  4  48 

6  2-88 

3-63 

0-92 

22  4  48 

13  14-88 

7-95 

0-43 

10  19  12 

6  11-52 

3-72 

0-93 

22  19  12 

13  23-52 

8-04 

0-44 

10  33  36 

6  20-16 

3-80 

0-94 

22  33  36 

13  32-16 

8-12 

0-45 

10  48  0 

6  28-80 

3-89 

0-95 

22  48  0 

13  40-80 

8-21 

0-46 

11  2  24 

6  37-44 

3-97 

0-96 

23   2  24 

13  49-44 

8-29 

0-47 

11  16  48 

6  46-08 

4-06 

0-97 

23  16  48 

13  58-08 

8-38 

0-48 

11  31  12 

6  54-72 

4-15 

0-98 

23  31  12 

14  6-72 

8-47 

0-49 

11  45  36 

7  3-36 

4-23 

0-99 

23  45  36 

14  15-36 

8-55 

0'50 

12   0  0 

7  12-00 

4-32 

1-00 

24  0   0 

14  24-00 

8-64 

APPENDIX   II. 


403 


TABLE   V. 

Greenwich  Mean  Time  of  the  Beginning  of  the  Adopted  Solar  Year  from 
1900  to  2000. — Mean  Longitude  of  the  Moon's  Node  and  Perigee. — 
Moon's  Mean  Longitude. 


Year. 

Begin- 
ning of 
Solar 
Year. 

Long1. 

d's 
Node. 

n 

Long. 
d's 
Perigee. 

n 

(Ts 
Mean 
Longi- 
tude. 

Year. 

Begin- 
ning of 
Solar 
Year. 

Long. 

<Ts 
Node. 

n 

Long. 

<TS 

Perigee. 

n 

r« 

Mean 
Longi- 
tude. 

Jan. 

~» 

0 

o 

Jan. 

o 

0 

0 

1900 

0-313 

259-16 

334-4 

274-6 

1950 

0-423 

12-11 

208-8 

63-4 

01 

0-556 

239-82 

15-0   47-1 

51 

0-666 

352-77 

249-5 

196-0 

02 

0-798 

220-48 

55-7  1  179-7 

1952B 

0-908 

333-43 

290-2 

328-5 

03 

1-040 

201-14 

96-4  312-3 

53 

0-150 

314-09 

330-9 

101-1 

1904s 

1  -282 

181-80 

137-1   84-9 

54 

0-392 

294-75 

11-6 

233-7 

05 

0-524 

162-46 

177-8  217-5 

55 

0-634 

275-41 

52-3 

6-3 

06 

0-767 

143-12 

218-5 

350-0 

1956B 

0-877 

256-07 

93-0 

138-8 

07 

1-009 

123-78 

259-2 

122-6 

57 

0-119 

236-73 

133-7 

271-4 

1908B 

1-251 

104-44 

299-9 

255-2 

58 

0-361 

217-38 

174-4 

44-0 

09 

0-493 

85-09 

340-6 

27-8 

59 

0-603 

198-04 

215-0 

176-6 

10 

0-735 

65-75 

21-3 

160-3 

1960s 

0-845 

178-70 

255-7 

309-1 

11 

0-978 

46-41 

61-9 

292-9 

61 

0-088 

159-36 

296-4 

81-7 

1912B 

1-220 

27-07 

102-6 

65-5 

62 

0-330 

140-02 

337-1 

214-3 

13 

0-462 

7-73 

143-3 

198-1 

63 

0-572 

120-68 

17-8 

346-9 

14 

0-704 

348-39 

184-0 

3306 

1964B 

0-814 

101  -34 

58-5 

119-4 

15 

0-946 

329-05 

224-7 

103-2 

65 

0-056 

82-00 

99-2 

252-0 

1916B 

1-189 

309-71 

265-4 

235-8 

66 

0-299 

62-66 

139-9 

24-6 

17 

0-431 

290-37 

306-1 

8-4 

67 

0-541 

43-32 

180-6 

157-2 

18 

0-673 

271-03 

346-8 

140-9 

1968B 

0-7S3 

23-97 

221-3 

289-8 

19 

0-915 

251-68 

27-5 

273-5 

69 

0-025 

4-63 

261-9 

62-3 

1920B 

1-157 

232-34 

68-2 

46-1 

70 

0-267 

345-29 

302-6 

194-9 

21 

0-400 

213-00 

108-8 

178-7 

71 

0-510 

325-95 

343-3 

327-5 

22 

0-642 

193-66 

149-5 

311-2 

1972B 

0-752 

306-61 

24-0 

100-1 

23 

0-884 

174-32 

190-2 

83-8 

73 

-0-006 

287-27 

64-7 

232-6 

1924B 

1-126 

154-98 

230-9 

216-4 

74 

0-236 

267-93 

105-4 

5-2 

25 

0-368 

135-64 

271-6 

349-0 

75 

0-478 

248-59 

146-1 

137-8 

26 

0-611 

116-30 

312-3 

121-6 

1976s 

0-720 

229-25 

186-8 

270-4 

27 

0-853 

96-96 

353-0 

254-1 

77 

-0-037 

209-91 

227-5 

42-9 

1928s 

1-095 

77-62 

33-7 

26-7 

78 

0-205 

190-56 

268-2 

175-5 

29 

0-337 

58-27 

74-4 

159-3 

79 

0-447 

171-22 

308-8 

308-1 

30 

0-579 

38-93 

115-0 

291-9 

1980B 

0-689 

151-88 

349-5 

80-7 

31 

0822 

19-59 

155-7 

64-4 

81 

-0-069 

132-54 

30-2 

213-2 

1932B 

1-064 

0-25 

196-4 

197-0 

82 

0-174 

113-20 

70-9 

345-8 

33 

0-306 

340-91 

237-1 

329-6 

83 

0-416 

93  -S6 

111*6 

118-4 

34 

0-548 

321-57 

277-8 

102-2 

1984B 

0-658 

74-52 

152-3 

251-0 

35 

0-790 

302-23 

318-5 

234-7 

85 

-0-100 

55-18 

193-0 

23-5 

1936s 

1-033 

282-89 

339-2 

7-3 

86 

0-142 

35-84 

233-7 

156-1 

37 

0-275 

263-55 

39-9 

139-9 

87 

0-385 

16-50 

274-4 

288-7 

38 

0-517 

244-21 

80-6 

272-5 

1988s 

0-6-27 

357-15 

315-0 

61-3 

39 

0-759 

224-86 

121-3 

45-0 

89 

-0-131 

337-81 

355-7 

193-9 

1940e 

1-001 

205-52 

161-9 

177-6 

90 

0-111 

318-47 

36-4 

326-4 

41 

0-244 

186-18 

202-6 

310-2 

91 

0-353 

299-13 

77-1 

99-0 

42 

0-486 

166-84 

243-3 

82-8 

1992B 

0-596 

279-79 

117-8 

231-6 

43 

0-728 

147-50 

284-0 

215-3 

93 

-0-162 

260-45 

158-5 

4-2 

1944B 

0-970 

128-16 

324-7 

347-9 

94 

0-080 

241-11 

199-2 

136-7 

45 

0-212 

108  82 

5-4 

120-5 

95 

0-322 

221-77 

239-9 

269-3 

46 

0-455 

89-48 

46-1 

253-1 

1996B 

0-564 

202-43 

280-6 

41-9 

47 

0-697 

70-14 

86-8 

25-7 

97 

-0-193 

183-09 

321-3 

174-5 

48B 

0-939 

50-80 

127-5 

158-2 

98 

0-049 

163-74 

1-9 

307-0 

49 

0-181 

31*45 

168-2 

290-8 

99 

0-291 

144-40 

42-6 

79-6 

1950 

0-423 

12-11 

208-8 

63-4 

2000B 

0-533 

125-06 

83-3 

212-2 

404 


APPENDIX   II. 


TABLE   VI. 

Motions  of  Moon's  Node,  Perigee,  and  Mean  Longitude. 


Sidereal 
Days. 

Solar  Date. 

Motion  of 

n                    n                      i 

U 

Jan.        O'O 

-o-ooo        +  o-oo 

+    o-oo 

10 

10-0 

0-528 

I'll 

131-40 

20 

19-9 

1-056 

222 

262-81 

30 

29-9 

1-584 

3-33 

34-21 

40 

Feb.       8-9 

2-112 

4-44 

165-62 

50 

18-9 

2-640 

5-55 

297-02 

60 

28-8 

3-169 

6-67 

68-43 

70 

Mar.    10-8* 

3-697 

7-78 

199-83 

80 

20-8* 

4-225 

8-89 

331-24 

90 

30-8* 

4-753 

10-00 

102-64 

100 

Apr.      9-7* 

5-281 

11-11 

234-04 

110 

19-7* 

5-809 

12-22 

5-45 

120 

29-7* 

6337 

1333 

136-85 

130 

May      9-6* 

6-865 

14-44 

268-26 

140 

19-6* 

7-393 

15-55 

39-66 

150 

29-6* 

7-921 

16-66 

171-07 

160 

June      86* 

8-450 

17-78 

302-47 

170 

18-5* 

8-978 

18-89 

73-88 

ISO 

28-5* 

9-506 

2000 

205-28 

190 

July      8-5* 

10-034 

21-11 

336-68 

200 

18-5* 

10-562 

22-22 

108-09 

210 

28-4* 

1  1  -090 

23-33 

239-49 

220 

Aug.      7'4* 

11-618 

24-44 

10-90 

230 

17'4* 

12-146 

25  -55 

142-30 

240 

27-3* 

12-674 

26-66 

273-71 

250 

Sept.     6-3* 

13-202 

27-77 

45-11 

2<iO 

16-3* 

13-730 

28-89 

176-52 

270 

26-3* 

14-259 

30-00 

307-92 

280 

Oct.       6'2* 

14-787 

31-11 

79-33 

290 

16-2* 

15-315 

32-22 

210-73 

300 

26-2* 

15843 

33-33 

342-13 

310 

Nov.      5-2* 

16-371 

34-44 

113-54 

320 

151* 

16-899 

35  -55 

244-94 

330 

25-1* 

17-427 

36-66 

16-35 

340 

Dec.       5-1* 

17-955 

37  77 

147-75 

350 

15-0* 

18-483 

38-88 

279-16 

360 

25-0* 

19-011 

40-00 

50-56 

370 

35-0* 

-19-540 

+  41-11 

+  181-97 

In  Bissextile  years,  the  dates  after  March  1  are  to  be  diminished  one  day. 


APPENDIX   II. 


405 


TABLE  VII. 

Motion  of  Moon's  Mean  Longitude  for  Tenths  of  a  Day. 


Days 
(Sidereal). 

Motion  of 

d 

Days 
(Sidereal). 

Motion  of 

d 

0 

0 

o-o 

o-oo 

5-0 

65-70 

o-i 

1-31 

.VI 

67-02 

0-2 

2-<)3 

5-2 

68-33 

0-3 

3-94 

5-3 

69-64 

0-4 

5-26 

5-4 

70-96 

0-5 

6-57 

5'5 

72-27 

0-6 

7-88 

5-6 

73-59 

0-7 

9-20 

5-7 

74-90 

0-8 

10-51 

5-8 

76-21 

0-9 

1  1  -83 

5-9 

77-53 

1-0 

13-14 

6-0 

78-84 

1-1 

14-45 

6-1 

80-16 

1-2 

15-77 

6-2 

81-47 

14 

17-08 

6-3 

82-79 

14 

18-40 

6-4 

84-10 

1-5 

19-71 

6-5 

85-41 

1-6 

21-02 

6-6 

86-73 

T7 

22-34 

6-7 

88-04 

1-8 

23-65 

6-8 

89-36 

1-9 

24-97 

6-9 

90-67 

2-0 

26-28 

7-0 

91-98 

2-1 

27  60 

7-1 

93-30 

2-2 

28-91 

7-2 

94-61 

2-3 

30-22 

7-3 

95-93 

2'4 

31-54 

7-4 

97-24 

2-5 

3-2-So 

7-5 

98-55 

2-6 

34-17 

7-6 

99-87 

2-7 

35-48 

7-7 

101-18 

2-8 

36-79 

7-8 

102-50 

2-9 

38-11 

7-9 

103-81 

3-0 

39-42 

8-0 

105-12 

3-1 

40-74 

8-1 

106-44 

3-2 

42-05 

8-2 

107-75 

3-3 

43-36 

8-3 

109-07 

3-4 

44-68 

8-4 

110-38 

3-5 

45-99 

8-5 

111-69 

3-6 

47-31 

8-6 

113-01 

3-7 

48-62 

8-7 

114-32 

3-8 

49-93 

8-8 

115-64 

3-9 

51-25 

8-9 

116-95 

4-0 

52-56 

9-0 

118-26 

4-1 

53-88 

9-1 

119-58 

4-2 

55-19 

9-2 

120-89 

4-3 

56-50 

9-3 

122-21 

4-4 

57-82 

9-4 

123-52 

4-5 

59-13 

9-5 

124-83 

4-6 

60-45 

9-6 

126-15 

4-7 

61-76 

9-7 

127  -4B 

4-8 

63-07 

9-8 

128-78 

4-9 

64-39 

9-9 

130-09 

5-0 

65-70 

10-0 

131-04 

APPENDIX   III. 


TABLE  VIII. 

Centennial  rates  of  the  precessional  motions  from  1750  to  2000. 


Motion  in  R.  A. 

Polar  Motion. 

me 

rcc 

log  nc                        nc                      log  nc 

s. 

s. 

H 

1750 

306-955 

133-731 

2-126232           2005-96 

3-302323 

1775 

307*001 

133717 

2-126  186           2005-75 

3-302277 

1800 

307-048 

133-703 

2-126140           2005-54 

3-302231 

1825 

307-094 

133-689 

2-126094           2005-32 

3-302  185 

1850 

307-141 

133-674 

2-126048           2005-11 

3-302  139 

1875 

307-187 

133-660 

2-126001           2004-90 

3-302092 

1900 

307-234 

133-646 

2-125955           2004-68 

3-302046 

1925 

307-280 

133-632 

2-125909 

2004-47 

3-302000 

1950 

307-327 

133-617 

2-125  863 

2004-26 

3-301954 

1975 

307-373 

133-603 

2-125817 

2004-04           3-301  908 

2000 

307-420 

133-589 

2-125771 

2003-83 

3-301862 

Date. 

Luui-solar 
Precession. 

General 
Precession. 

Precession  from  Motion  of  Ecliptic. 

in  R.A.,  X' 

in  Long.,  A'  cos  e 

P 

I 

1750 

5036"-34 

5022"  -30              15"  -30 

14"-03 

1775 

5036  -47 

5022  -86               14  -83                  13  -60 

1800 

5036  -59 

5023  -41 

14  -36 

13  -17 

1825 

5036  -71 

5023  -97 

13  -89 

12  -74 

1850 

5036  -84 

5024  -53 

13  -42                  12  -31 

1875 

5036  -96 

5025  -08 

12  -95 

11  -88 

1900 

5037  -08 

5025  -64 

12  -48 

11  -45 

1925 

5037  '21 

5026  -19 

12-00 

11  -02 

1950 

5037  -33 

5026  -75 

11   53 

10  -58 

1975 

5037  '45 

5027  -31 

11  -06 

10  -15 

2000 

5037  -58 

5027  -86 

10  -59 

9  -72 

Formulae  for  the  annual  precessions. 
In  R. A.,  pa  =  m  +  n  sin  oc  tan  S, 

In  Dec.,  ps  =  ncosoi, 

where  m  =  mc  -=- 100  ;  n  =  nc  -f  100. 

Formulae  for  the  centennial  precessions. 

(Same  as  for  the  annual  precessions,  using  mc  for  m  and  nc  for  n.) 
Centennial  variations  of  the  centennial  variations  of  a  and  8. 
a  +  A  (pac  +  2fjia)  cos  a 
+  £(jpsc  +  2/xs)  since 
+  [4-9866  -10]fw*«  tan  8, 
CS  -[9-1637  -  lOlQ^  +  fyJsina. 

-[6-7367  -10J/4  sin  28, 
where  /xa  is  the  centennial  proper  motion  of  a  in  seconds   of  time, 
P&  that  of  8  in  seconds  of  arc,  and  A,  B,  Ca  and  C$  are  to  be  taken 
from  the  following  tables.     (Compare  §  146.) 


APPENDIX   III. 


407 


TABLES  FOE  COMPUTING  THE  SECULAR  VARIATIONS  OF 
THE  CENTENNIAL  PROPER  MOTIONS  OF  THE  STARS. 

TABLE  IX. 


Dec. 

Log.  A. 

Log.  B. 

Dec. 

Log.  A. 

Log.  B. 

Dec. 

Log.  A. 

Log.  B. 

0      0 

6-8115 

7      0 

7-0767 

6-8180 

14      0 

7-3844 

6*8377 

V           v 

10 

5-4513 

•8115 

10 

•0871 

•8183 

10 

•3897 

•8383 

20 

5  7524 

•8115 

20 

•0972 

•8186 

20 

•3950 

•8390 

30 

5-9285 

•8115 

30 

•1070 

•8190 

30 

•4003 

•8396 

40 

6-0534 

•8116 

40 

•1167 

•8193 

40 

•4054 

•8403 

50 

•1503 

•8116 

50 

•1261 

•8196 

50 

•4106 

•8409 

1      0 

6-2295 

6-8116 

8      0 

7-1354 

6-8200 

15      0 

7-4157 

6-8416 

10 

•2965 

•8117 

10 

•1445 

•8204 

10 

•4207 

•8423 

20 

•3545 

•8117 

20 

•1534 

•8207 

20 

•4257 

•8429 

30 

•4057 

•8118 

30 

•1621 

•8211 

30 

•4306 

•8437 

40 

•4514 

•8119 

40 

•1707 

•8215 

40 

•4355 

•8444 

50 

•4929 

•8119 

50 

•1791 

•8219 

50 

•4403 

•8451 

2      0 

6-5307 

6-8120 

9      0 

7-1873 

6-8223 

16     0 

7-4451 

6-8458 

10 

•5655 

•8121 

10 

•1954 

•8227 

10 

•4498 

•8465 

20 

•5977 

•8122 

20 

•2034 

•8231 

20 

•4545 

•8473 

30 

•6277 

•8123 

30 

•2112 

•8235 

30 

•4592 

•8480 

40 

•6558 

•8124 

40 

•2189 

•8239 

40 

•4638 

•8488 

50 

•6821 

•8126 

50 

•2265 

•8244 

50 

•4684 

•8495 

3      0 

6-7070 

6-8127 

10      0 

7-2339 

6-8248 

17      0 

7-4729 

6-8503 

10 

•7305 

•8128 

10 

•2412 

•8252 

10 

•4774 

•8511 

20 

•7528 

•8130 

20 

•2485 

•8257 

20 

•4819 

•8519 

30 

•7741 

•8131 

30 

•2556 

•8262 

30 

•4863 

•8527 

40 

•7943 

•8133 

40 

•2626 

•8266 

40 

•4907 

•8535 

50 

•8137 

•8134 

50 

•2695 

•8271 

50 

•4951 

•8543 

4      0 

6-8322 

6-8136 

11      0 

7-2763 

6-8276 

18      0 

7  '4994 

6-8551 

10 

•8500 

•8138 

10 

•2829 

•8281 

10 

•5037 

•8559 

20 

•8671 

•8140 

20 

•2896 

•8286 

20 

•5079 

•8567 

30 

•8836 

•8142 

30 

•2961 

•8291 

30 

•5121 

•8576 

40 

•8994 

•8144 

40 

•3025 

•8296 

40 

•5163 

•8584 

50 

•9148 

•8146 

50 

•3088 

•8302 

50 

•5205 

•8593 

5      0 

6-9296 

6-8148 

12      0 

7-3151 

6-8307 

19      0 

7-5246 

6-8602 

10 

•9439 

•8150 

10 

•3212 

•8312 

10 

•5287 

•8610 

20 

•9577 

•8153 

20 

•3273 

•8318 

20 

•5327 

•8619 

30 

•9712 

•8155 

30 

•3334 

•8323 

30 

•5367 

•8628 

40 

•9842 

•8158 

40 

•3393 

•8329 

40 

•5407 

•8637 

50 

6-9969 

•8160 

50 

•3452 

•8335 

50 

•5447 

•8646 

6     0 

7-0092 

6-8163 

13      0 

7-3510 

6-8341 

20      0 

7-5487 

6-8655 

10 

•0212 

•8165 

10 

•3567 

•8346 

10 

•5526 

•8665 

20 

•0329 

•8168 

20 

•3624 

•8352 

20 

•5565 

•8674 

30 

•0443 

•8171 

30 

•3680 

•8358 

30 

•5603 

•8683 

40 

•0554 

•8174 

40 

•3735 

•8364 

40 

•5642 

•8693 

50 

•0662 

•8177 

50 

•3790 

•8371 

50 

•5680 

•8703 

7      0 

7-0767 

6-8180 

14      0 

7-3844 

6-8377 

21      0 

7-5718 

6-8712 

408 


APPENDIX   III. 
TABLE  IX.  —  Continued. 


Dec. 

Log.  A. 

Log.  B. 

Dec. 

Log.  A. 

Log.  B. 

Dec. 

Log.  A. 

Log.  B. 

21      0 

7-5718 

6-8712 

29      0 

7-7314 

6-9279 

37      0 

7-8647 

7-0068 

10 

•5755 

•8722 

10 

•7343 

•9293 

10 

•8673 

•0087 

20 

•5793 

•8732 

20 

•7o73 

•9307 

20 

•8700 

•0106 

30 

•5830 

•8741 

30 

•7402 

•93-_>l 

30 

•8726 

•0126 

40 

•5867 

•8751 

40 

•7432 

•9335 

40 

•8752 

•0145 

50 

•5904 

•8762 

50 

•7461 

•9350 

50 

•8778 

•0165 

22      0 

7-5940 

6-8772 

30      0 

7  7490 

6-9364 

38      0 

7-8804 

7-01S4 

10 

•5976 

•8782 

10 

•7520 

•9379 

10 

•8830 

•0204 

20 

•6012 

•8792 

20 

•7549 

•9394 

20 

8856 

•0224 

30 

•6048 

•8803 

30 

•7577 

•9409 

30 

•8882 

•0244 

40 

•6084 

•8813 

40 

•7606 

•9424 

40 

•8908 

•0264 

50 

•6119 

•8824 

50 

•7635 

•9439 

50 

•8934 

•0285 

23      0 

7-6155 

6-8834 

31      0 

77664 

6-9454 

39      0 

7-8960 

7-0305 

10 

•6190 

•8845 

10 

•7692 

9469 

10 

•8986 

•0325 

20 

•6224 

•8856 

20 

•7721 

•9484 

20 

•9011 

0346 

30 

•6259 

•8867 

30 

•7749 

•9500 

30 

•9037 

•0367 

40 

•6293 

•8878 

40 

•7778 

•9515 

40 

•9063 

•0388 

50 

•6328 

•8889 

50 

•7806 

•9531 

50 

•9088 

•0409 

24      0 

7-6362 

6-8900 

32      0 

7*7834 

6-9547 

40      0 

7-9114 

7-0430 

10 

•6396 

•8912 

10 

•7862 

•9562 

10 

•9140 

•0451 

20 

•6429 

•8923 

20 

•7890 

•9578 

20 

•9165 

0473 

30 

•6463 

•8935 

30 

•7918 

•9594 

30 

•9191 

•0494 

40 

•6496 

•8946 

40 

•7946 

•9611 

40 

•9217 

•0516 

50 

•6530 

•8958 

50 

;7973 

•9627 

50 

•9242 

•0538 

25      0 

7-6563 

6-8969 

33      0 

7-8001 

6-9643 

41      0 

7-9268 

7-0559 

10 

•6596 

•8981 

10 

•8029 

•9660 

10 

•9293 

•0581 

20 

•6628 

•8993 

20 

•8056 

•9676 

20 

•9310 

•0(504 

30 

•6661 

•9005 

30 

•8084 

•9693 

30 

•9344 

•0626 

40 

•6693 

•9017 

40 

•8111 

•9710 

40 

•9370 

•0648 

50 

6726 

•9030 

50 

•8139 

•9727 

50 

•9395 

•0671 

26      0 

7-6758 

6-9042 

34      0 

7-8166 

6-9744 

42      0 

7'9420 

7-0694 

10 

•6790 

•9054 

10 

•8193 

•9761 

10 

•944G 

•0716 

20 

•6822 

•9067 

20 

•8220 

•9778 

20 

•9471 

•0739 

30 

•6853 

•9079 

iO 

•8247 

•9795 

30 

•9497 

•0762 

40 

•6885 

•9092 

40 

•8274 

•9813 

40 

•9522 

•0786 

50 

•6916 

•9105 

50 

•8301 

•9830 

50 

•9547 

•0809 

27      0 

7-6948 

6-9117 

35     0 

7-8328 

6-9848 

43      0 

7-9573 

7-0832 

10 

•6979 

•9130 

10 

•8355 

•9865 

10 

•9598 

•0856 

20 

•7010 

•9143 

20 

•8382 

•9883 

20 

•9623 

•0880 

30 

•7041 

•9156 

30 

•8409 

•9901 

30 

•9648 

•0904 

40 

•7072 

•9170 

40 

•8435 

•9919 

40 

•9674 

•0928 

50 

•7102 

•9183 

50 

•8462 

•9938 

50 

•9699 

•0952 

28      0 

7-7133 

6-9196 

36      0 

7-8489 

6-9956 

44      0 

7-9724 

7-0976 

10 

7163 

•9210 

10 

•8515 

•9974 

10 

•9750 

•1001 

20 

•7193 

•9223 

20 

•8542 

6-9993 

20 

•9775 

•1025 

30 

•7224 

•9237 

30 

•8568 

7-0011 

30 

•9800 

•1050 

40 

•7254 

•9251 

40 

•8594 

•0030 

40 

•9825 

•1075 

50 

•7284 

•9265 

50 

•8621 

•0049 

50 

•9851 

•1100 

29      0 

7'7314     6-9279 

37      0 

7-8647 

7-0068 

45      0 

7-9876 

7-1125 

APPENDIX  111. 
TABLE  IX.— Continued. 


409 


Dec. 

Log.  A. 

Log.  B. 

Dec. 

Log.  A. 

Log.  B. 

Dec. 

Log.  A. 

Log.  B. 

45      0 

7-9876 

7-1125 

53      0 

8-1105 

7-2526 

61      0 

8-2438 

7-4404 

10 

•9901 

•1151 

10 

•1131 

•2559 

10 

•2468        -4449 

20 

•9927 

•1176 

20 

•1158 

•2593 

20 

•2498        -4495 

30 

•9952 

•1202 

30 

•1184 

•2627 

30 

•2528  i      -4542 

40 

7-9977 

•1228 

40 

•1210 

•2661 

40 

•2559  i      -4588 

50 

8-0002 

•1253 

50 

•1237 

2696 

50 

•2589 

•4635 

46      0 

8-0028 

7-1280 

54      0 

8-1263 

7-2731 

62      0 

8-2619 

7-4683 

10 

•0053 

•1306 

10 

•1290 

•2766 

10 

•2650 

•4731 

20 

•0078 

•1332 

20 

•1317 

•2801 

20 

•2680 

•4779 

30 

•0104 

•1359 

30 

•1343 

•'2836 

30 

•2711 

•4827 

40 

•0129 

•1385 

40 

•1370 

•2871 

40 

•2742 

•4876 

50 

•0154 

•1412 

50 

•1397 

•2907 

50 

•2773 

•4925 

47      0 

8-0179  !   7-1439 

55      0 

8-1424 

7-2943 

63      0 

8-2804 

7-4974 

10 

•0205 

•1467 

10 

•1451 

•2979 

10 

•2836        -5024 

20 

•0230 

•1494 

20 

•1478 

•3016 

20 

•2867  1      '5074 

30 

•0255 

•1521 

30 

•1505 

•3052 

30 

•2899        '5124 

40 

•0281 

•1549 

40 

1532 

•3089 

40 

•2930        -5175 

50 

•0306 

•1577 

50 

•1559 

•3126 

50 

•2962        -5227 

48      0 

8-0332 

7-1605 

56      0 

8-1586 

7-3164 

64      0 

8-2994      7-5278 

10 

•0357 

•1633 

10 

•1613 

•3201 

10 

•3026        '5330 

20 

•0382 

•1661 

20 

•1641 

•3-239 

20 

•3059        '5383 

30 

•0408 

•1689 

30 

•1668 

•3277 

30 

•3091 

•5435 

40 

•0433 

•1718 

40 

•1696 

•3316 

40 

•3124 

•5488 

50 

•0459 

•1747 

50 

•1723 

•3354 

50 

•3156 

•5542 

49      0 

8-0484 

7-1776 

57      0 

8-1751 

7-3393 

65      0 

8-3189 

7-5596 

10 

•0510 

•1805 

10 

•1779 

•3432 

10 

•3222 

•5650 

20 

•0535 

•1835 

20 

•1806 

•3471 

20 

•3256 

•5705 

30 

•0561 

•1864 

30 

•1834 

•3511 

30 

•3289 

•5760 

40 

•0587 

•1894 

40 

•1862 

•3550 

40 

•3323  !      -5816 

50 

•0612 

•1923 

50 

•1890 

•3591 

50 

•3356        '5872 

50      0 

8-0638 

7-1954 

58      0 

8-1918 

7-3631 

66      0 

8-3390  :   7-5929 

10 

•0664 

•1984 

10 

•1946 

•3671 

10 

•3424 

•5986 

20 

•0689 

•2014 

20 

•1974 

•3712 

•20 

•3459 

•6043 

30 

•0715 

•2045 

30 

•2003 

•3753 

30 

•3493 

•6101 

40 

•0741 

•2076 

40 

•2031 

•3795 

40 

•3528 

•6159 

50 

•0766 

•2106 

50 

•2060 

•3836 

50 

•3562 

•6218 

51     0 

8-0792 

7-2138 

59      0 

8-2088 

7-3878 

67      0 

8-3597 

7-6277 

10 

•0818 

•2169 

10 

•2117 

•3920 

10 

•3633 

•6337 

20 

•0844 

•2200 

20 

•2146 

•3963 

20 

•3668  i      -6397 

30 

•0870 

•2232 

30 

•2175 

•4006 

30 

•3704 

•6458 

40 

•0896 

•2264 

40 

2203 

•4049 

40 

•3740 

•6519 

50 

•0922 

•2296 

50 

•2232 

•4092 

50 

•3776 

•6581 

52      0 

8-0948 

7-2328 

60      0 

8-2262 

7-4136 

68      0 

8-3812 

7  -0643 

10 

•0974 

•2361 

10 

•2291 

•4180 

10 

•3848 

•6706 

20 

•1000 

•2393 

20 

•2320 

•4224 

20 

•3885 

•6770 

30 

•1026 

•2426 

30 

•2350 

•4268 

30 

•3922 

•6833 

40 

•1052 

•2459 

40 

•2379 

•4313 

40 

•3959 

•6898 

50 

•1079 

•2492 

50 

•2409 

•4358 

50 

•3997 

•6963 

53      0 

8-1105 

7-2526 

61      0 

8-2438 

7-4404 

69     0 

8-4034 

7-7028 

410 


APPENDIX   III. 
TABLE  IX.— Concluded. 


Dec. 

Log.  A. 

Log.  B. 

Dec. 

Log.  A. 

Log.  B. 

Dec. 

Log.  A. 

Log.  B. 

69     0 

8-4034 

7-7028 

0                     1 

76      0 

8-5908 

8-0441 

83     0 

8-8985 

8-6397 

10 

•4072 

•7094 

10 

•5962 

•0543 

10 

•9090 

•6605 

20 

•4110 

•7161 

20 

•6017 

•0647 

20 

•9198 

•6819 

30 

•4149 

•7228 

30 

•6072 

•0751 

30 

•9309 

•7038 

40 

•4187 

•7296 

40 

•6128 

•0857 

40 

•9423 

•7263 

50 

•4226 

•7365 

50 

•6185 

•0965 

50 

•9540 

•7493 

70      0 

8-4265 

7-7434 

77      0 

8-6242 

8-1073 

84      0 

8-9660 

8-7730 

10 

•4305 

•7504 

10 

•6300 

•1183 

10 

•9783 

•7974 

20 

•4345 

•7574 

20 

•6359 

•1295 

20 

8-9910 

•8225 

30 

•4385 

•7645 

30 

•6418 

•1408 

30 

9-0040 

•8484 

40 

•4425 

•7717 

40 

•6479 

•1523 

40 

•0175 

•8750 

50 

•4465 

•7789 

50 

•6540 

•1639 

50 

•0313 

•9025 

71      0 

8-4506 

7-7862 

78      0 

8-6601 

8-1757 

85      0 

9-0456 

8-9309 

10 

•4547 

•7936 

10 

•6664 

•1877 

10 

•0604 

•9603 

20 

•4589 

•8010 

20 

•6727 

•1999 

20 

•0758 

8-9907 

30 

•4631 

•8085 

30 

•6791 

•2122 

30 

•0916 

9-0222 

40 

•4673 

•8161 

40 

•6856 

•2247 

40 

•1081 

•0549 

50 

•4715 

•8238 

50 

•6923 

•2374 

50 

•1252 

•0889 

72      0 

8-4758 

7-8315 

79     0 

8-6989 

8-2503 

86      0 

9-1430 

9-1243 

10 

•4801 

•8394 

10 

•7057 

•2634 

10 

•1615 

•1612 

20 

•4845 

•8472 

20 

•7126 

•2767 

20 

•1809 

•1998 

30 

•4889 

•8552 

30 

•7196 

•2902 

30 

•2011 

•2401 

40 

•4933 

•8633 

40 

•7267 

•3040 

40 

•2224 

•2825 

50 

•4978 

•8714 

50 

•7340 

•3180 

50 

•2447 

•3270 

73      0 

8-5023 

7-8796 

80      0 

8-7413 

8-3322 

87      0 

9-2682 

9-3739 

10 

•5068 

•8879 

10 

•7487 

•3466 

10 

•2931 

•4235 

20 

•5114 

•8963 

20 

•7563 

•3613 

20 

•3194 

•4761 

30 

•5160 

•9048 

30 

•7640 

•3763 

30 

•3475 

•5321 

40 

•5207 

•9134 

40 

•7718 

•3915 

40 

•3775 

•5920 

50 

•5254 

•9221 

50 

•7798 

•4070 

50 

•4097 

•6564 

74      0 

8-5301 

7-9308 

81      0 

8-7879 

8-4228 

88      0 

9-4445 

9-7259 

10 

•5349 

•9397 

10 

•7961 

•4389 

10 

•4823 

•8014 

20 

•5397 

•9486 

20 

•8045 

•4554 

20 

•5238 

•8842 

30 

•5446 

•9577 

30 

•8131 

•4721 

30 

•5695 

9-9757 

40 

•5495 

•9669 

40 

•8218 

•4892 

40 

•6207 

0-0779 

50 

•5545 

•9761 

50 

•8307 

•5066 

50 

•6787 

•1939 

75      0 

8-5595 

7-9855 

82      0 

8-8398 

8-5244 

89      0 

9-7457 

0-3278 

10 

•5646 

•9950 

10 

•8491 

•5426 

10 

•8249 

•4861 

20 

•5698 

8-0046 

20 

•8585 

•5611 

20 

9-9218 

•6799 

30 

•5749 

•0143 

30 

•8682 

•5801 

30 

0-0467 

0-9298 

40 

•5802 

•0241 

40 

•8780 

•5995 

40 

•2228 

1  -2820 

50 

•5855 

•0341 

50 

•8881 

•6194 

50 

0-5239 

1-8840 

76      0 

8'5908 

8-0441 

83      0 

8-8985 

8-6397 

90      0 

APPENDIX   III. 
TABLE   X. 


411 


Ann.  Prec. 
in  R.A. 

Co.. 

Ann.  Prec. 
in  R.A. 

Ca. 

Ann.  Prec. 
in  Dec. 

Cs. 

-2:0 

+  0-402 

+  3-0 

+  0490 

-20 

+  0-85 

-1-9 

•398 

3-1 

•185 

-19 

•81 

-1-8 

•394 

3-2 

•181 

-18 

•77 

-1-7 

+  0-389 

3'3 

+  0-177 

-17 

+  0-73 

-1-6 

•385 

3-4 

•173 

-16 

•68 

-1-5 

•381 

3-5 

•168 

-15 

•64 

-1-4 

+  0-377 

3-6 

+  0-164 

-14 

+  0-60 

-1-3 

•372 

3-7 

•160 

-13 

•56 

-1-2 

•368 

3-8 

•156 

-12 

•51 

-1-1 

+  0-364 

3-9 

+  0-151 

-11 

+  0-47 

-1-0 

•360 

4-0 

•147 

-10 

•43 

-0-9 

•355 

4-1 

•143 

-    9 

•39 

-0-8 

+  0-351 

4-2 

+  0-139 

-   8 

+  0-34 

-0-7 

•347 

4-3 

•134 

_    n 

•30 

-0-6 

•343 

..        4-4 

•130 

-   6 

•26 

-0-5 

+  0-338 

4-5 

+  0-126 

-   5 

+  0-22 

-0-4 

•334 

4-6 

•122 

-   4 

•17 

-0-3 

•330 

4-7 

•117 

-   3 

•13 

-  0"2 

+  0-326 

4-8 

+  0-113 

-   2 

+  0-09 

-o-i 

•321 

4-9 

•109 

-    1 

•04 

o-o 

•317 

5-0 

•105 

0 

•00 

+0-1 

+  0-313 

5-1 

+  0-100 

+   1 

-0-04 

0-2 

•309 

5-2 

•096 

2 

•09 

0-3 

•304 

5-3 

•092 

3 

•13 

0-4 

+  0-300 

5-4 

+  0-088 

4 

-0-17 

0-5 

•296 

5-5 

•083 

5 

•22 

0-6 

•292 

5-6 

•079 

6 

•26 

0-7 

+  0-287 

5-7 

+  0-075 

7 

-0-30 

0-8 

•283 

5-8 

•071 

8 

•34 

0-9 

•279 

5-9 

•066 

9 

•39 

1-0 

+  0-275 

6-0 

+  0-062 

10 

-0-43 

1-1 

•270 

6-1 

•058 

11 

•47 

1-2 

•266 

6'2 

•054 

12 

•51 

1-3 

+  0-262 

6-3 

+  0-049 

13 

-0-56 

1-4 

•258 

6-4 

•045 

14 

•60 

1-5 

•253 

6-5 

•041 

15 

•64 

1-6 

+  0-249 

6-6 

+  0-037 

16 

-0-68 

1-7 

•245 

6-7 

•032 

17 

•73 

1-8 

•241 

6-8 

•028 

18- 

•77 

1-9 

+  0-236 

6-9 

+  0-024 

19 

-0-81 

2-0 

•232 

7-0 

•020 

+  20 

-0-85 

2-1 

•228 

7-1 

•015 

y    2-2 

+  0-224 

7-2 

+0-011 

2-3 

•219 

7'3 

•007 

2-4 

•215 

7-4 

+  0-003 

2-5 

+  0-211 

7'5 

-0-002 

2-6 

•207 

7'6 

-    -006 

2-7 

•202 

7-7 

-  -010 

2-8 

+  0-198 

7-8 

--  0-014 

2-9 

•194 

7'9 

-   -019 

+  3-0 

+  0-190 

+  8-0 

-0-023 

APPENDIX   IV. 

CONSTANTS  AND   TABLES   FOR  THE  TRIGONOMETRIC 
REDUCTION   OF   MEAN   PLACES   OF  THE   STARS. 

I.— General  Expressions  for  the  Constants  of  Reduction. 

The  constants  f0,  zt  and  6,  of  which  the  general  expressions  follow, 
fix  the  position  of  the  mean  equator  and  equinox  at  an  epoch  TQ  +  T 
relative  to  their  positions  at  an  initial  epoch  TQ.  T  is  the  interval 
between  the  two  epochs,  expressed  in  terms  of  100  solar  years,  or 
36524-22  days,  as  the  unit  of  time. 

The  geometric  meaning  of  these  constants  may  be  gathered  from 
the  chapter  on  Precession,  §§  127-131,  in  which  they  are  developed. 
The  expressions  for  initial  epochs  intermediate  between  those  given 
can  be  found  by  interpolation. 

In  using  the  numbers  f0,  z,  and  0  to  reduce  mean  places  of  stars, 
the  latter  are  supposed  to  be  given  for  the  initial  epoch,  and  the 
problem  is  to  reduce  them  to  the  epoch  T0  +  T,  which  may  be  earlier 
or  later.  For  cases  when  the  initial  epoch  is  earlier  than  1850,  the 
general  expressions  may  be  extended  by  carrying  the  coefficients 
of  T  and  its  powers  backward  by  means  of  the  uniform  centennial 
change  derived  from  the  given  coefficients. 

We  may  also,  in  any  case,  form  the  constants  by  the  principle 
that  the  two  epochs  are  interchangeable,  provided  that  we  also  change 

to,  z,  and  0 
into  -  z,   -  (0,  and  -  0. 

For  example,  if  we  wish  to  reduce  the  positions  of  the  Bradley 
stars  from  1755  to  1875,  we  may  form  the  numbers  for  the  reverse 
reduction  from  1875  as  the  initial  epoch  to  1755. 


APPENDIX   IV.  413 


Putting  T=  -1-20, 

we  thus  find  f0=  -  2764-28, 

z=  -2763-14, 
0=  -2406-42. 

Hence,  for  reducing  from  1755  to  1875,  we  use 
C0  =  2763-14, 


(9  =  2406-42. 

To  obtain  the  numbers  without  this  reversal,  we  find  that  in  £0 
coefficient  of  T  is  represented  by  the  expression 

2303"-55  +  l"-40(ro-1850). 
We  therefore  have,  when  jT0=  1755, 

&  =  2302"-227'  +  (T-30712  +  0"'0l7r3, 
which  gives,  for  T=  1-20, 


1  -14 


3  =  2764  -27 

It  will  be  seen  that  these  numbers  agree  with  those  found  by 
reversal  within  0"'01. 

This  relation  between  the  numbers  for  the  two  epochs  affords  a 
check  upon  the  correctness  of  the  expressions  as  printed,  because,  for 
example,  the  numbers  found  for  770  =  1850,  T=  +  1  should  correspond 
to  those  found  for  TQ  =  1950  ;  T=  -  1.  In  fact,  we  find, 

1850  to  1950.  1950  to  1850. 

C0  =  2303-87.  &  =  -2304-67. 

3=2304-66.  z=  -2303-88. 

6  =  2004-64.  B=  -  2004-65. 

The  coefficients  of  T  and  its  powers  in  all  the  expressions  increase 
or  diminish  uniformly  with  the  time  for  several  centuries.  This 
check  can,  therefore,  alwajs  be  applied  by  continuing  the  expressions 
to  former  values  of  the  initial  epoch  by  addition  or  subtraction. 

It  will  be  noted  that  these  three  constants  are  always  positive  when 
the  initial  epoch  is  the  earlier  of  the  two,  and  negative  when  it  is 
the  later. 


414  APPENDIX   IV. 


General  Expressions  for  the  Constants  of  Reduction. 

Initial  epoch.  Expression. 

n 

1850  f0  =  2303--5571  +  Q"'30T*  +  (T-017773 

1875  2303-90    +0-30      +0-017 

1900  2304-25    +0-30      +0*017 

1925  2304-60    +0  -3Q     +0-017 

1950  2304-95    +0-30      +0  -017 

1850  6>=2005"-1177-0//-43772-0"-04ir3 

1875  2004-90    -0-43      -0*041 

1900  2004-68    -0-43      -0-041 

1925  2004-47    -0'43      -0-041 

1950  2004-26    -0*43      -0-041 


1850  V/ 
1875  5136  -95  -1-07  -0  -001 
1900  5137  -07  -1-07  -0  -001 
1925  5137  -20  -  1  -07  -0  -001 
1950  5137  -32  -  1  -07  -0  -001 

1850  A  =  13"-4271  -  2"-38r2  -  0"'003r3 
1875         12  -95  -2-38   -0  -003 
1900         12  -48'  -2-38   -0  -003 
1925         12-00  -2-38   -0  -003 
1950         11-53  -2-38   -0  -003 


APPENDIX   IV. 


415 


II.— Special  Values  for  the  Usual  Epochs. 

During  the  present  generation  the  common  equinox  to  which  all 
star  positions  are  reduced  for  the  purpose  of  comparison  will  generally 
be  either  1875  or  1900.  We  therefore  give  tables  of  the  special 
values  of  £0,  s,  0,  and  m,  as  derived  from  the  preceding  expressions, 
for  reduction  from  the  dates  of  the  principal  catalogues  of  stars  to 
the  equinoxes  of  these  two  epochs. 


Special  Values  of  Constants  foi*  the  Reduction  of  Mean  Places  of  the  Stars 
from  various  dates  to  the  Equinox  and  Equator  0/1875  and  1900. 

TABLE    XlA,    FOR   REDUCTION   TO    1875. 


Date. 

it- 

z. 

e. 

log  h  sin  6. 

m. 

1755 

46      3-14 

46       4-28 

40      6-92 

2-20527 

m.          s. 
+  6      8-495 

1800 

28    47-32 

28    47-76 

25      3-84 

2-001  12 

3    50-339 

1825 

19     11-68 

19     11-88 

16    42-55 

1-82501 

2    33-571 

1830 

17     16-54 

17     16-70 

15      2-29 

1  -779  25 

2     18-216 

1840 

13    26-23 

13    26-33 

11     41-76 

1-67010 

1     47-504 

1845 

11     31-07 

11     31-14 

10       1  -51 

1-60315 

1     32-147 

1850 

9    35-91 

9     35-96 

8    21-25 

1-52396 

1     16-791 

1855 

7    4074 

7    40-77 

6    41-00 

1-42704 

1       1  -434 

1860 

5    45-56 

5    45-58 

5      0-74 

1-30211 

0    46-076 

1865 

3    50-38 

3    50-39 

3    20-49 

1-12601 

0    30-718 

1870 

1     55-19 

1     55-19 

1     40-25 

0-824  98 

,    0     15-359 

1875 

0      0-00 

0      0-00 

0      0-00 

o    o-ooo 

1880 

-    1     55-20 

-    1     55-20 

-    1     40-24 

0-82496,, 

-0     15-360 

1885 

-   3    50-40 

-   3    50-39 

-    3    20-48 

1-12599B 

-0    30-720 

1890 

-   5    45-61 

-   5    45-59 

-   5      0-72 

1  -302  08n 

-0    46-080 

1895 

-   7     40-82 

-   7    40-79 

-   6    40-96 

1-427  Oln 

-1       1-441 

1900 

-   9    36-04 

-   9    35-99 

-   8    21-20 

l-52392n 

-1     16-802 

1905 

-11     31-27 

-11     31-20 

-  10       1  -43 

l-60309n 

-1     32-164 

1910 

-13    26-50 

-13    26-40 

-11     41-66 

1-67004M 

-1     47-527 

1915 

-15    21-74 

-15    21-61 

-13    21-89 

1  -728  02n 

-2      2-890 

1920 

-17     16-98 

-17     16-82 

-15      2-11 

1  -779  17n 

-2     18-253 

1925 

-  19     12-22 

-  19     12-03 

-16    42-34 

1-82492,, 

-2    33-617 

1930 

-21      7-48 

-21      7-24 

-18    22-56 

l-8«631n 

-2    48-981 

1935 

-23      2-74 

-  23      2-45 

-20      2-78 

l-90409n 

-3      4-346 

1940 

-24    58-00 

-24    57-67 

-21     42-99 

l-95885n 

-3     19-711 

1945 

-26    5327 

-26    52-88 

-23    23-20 

1-97103M 

-3    35-077 

1950 

-28    48-55 

-28    48-10 

-25      3-42 

2-00098n 

-3    50-443 

416 


APPENDIX   IV. 


TABLE   XlB.,    FOR   REDUCTION   TO    1900. 


Date. 

& 

z. 

e. 

log  h  sin  d. 

m. 

1755 

55    38-92 

55    40-58 

48    27-56 

2-287  42 

m.         s. 
+  7     25-300 

1800 

38     23-18 

38    23-96 

33    25-06 

2-12603 

5      7-143 

1825 

28    47-58          28    48-02 

25      3-73 

2-001  08 

3    50-373 

18.30 

26    52-45 

26    52-83 

23    23-47 

1-971  11 

3    35-019 

1835 

24    57-31 

24    57-64 

21     43-21 

1-93892 

3     19-663 

1840 

23      2-16 

23      2-44 

•20      2-95 

1-90416 

3      4-307 

1845 

21       7-01 

21       7-25 

18     22-70 

1-86636 

2     48-951 

1850 

19     11-85 

19     12-05 

16    42-44 

1-82497 

2    33-593 

1855 

17     16-69 

17     16-85 

15      2-19 

1-77921 

2     18-236 

1860 

15    21-53 

15    21-05 

13    21-94 

1-72805 

2      2-879 

1865 

13    26-35 

13    26-45 

11     41-69 

1-67005 

1     47-520 

1870 

11     31-18 

11     31-25 

10       1-44 

1-60310 

1     32-162 

1875 

9    35-99 

9    36-04 

8     21-20 

1-52392 

1     16-802 

1880 

7    40-81 

7    40-84 

6     40-95 

1  -427  00 

1       1-443 

1885 

5    45-61 

5    45-63 

5      0-71 

1  -302  06 

0    46-083 

1890 

3    50-41 

3    50-42 

3    20-47 

1-12596 

0    30-722 

1895 

1     55-21 

1     55-21 

1     40-24 

0-824  93 

0     15-361 

1900 

o    o-oo 

o    o-oo 

o    o-oo 

o     o-ooo 

1905 

-    1     55-21 

-    1     55-22 

-    1     40-23 

0-824  92n 

-0     15-362 

1910 

-    3    50-44         -   3     50-43 

-   3    20-46 

l-12594n 

-0    30-725 

1915 

-   5     45-66        -   5    45-64 

-   5      0-69 

l-30203n 

-0     46-087 

1920 

-   7    40-89     !   -   7    40-86 

-   6    40-92 

l-42696n 

-1       1-450 

1925 

-   9    36-13     I   -   9    36-08 

-    8    21-14 

l-52387n 

-1     16-814 

1930 

-11     31-37       -11     31-30 

-10       1-36 

l-60304n 

-  1     32-178 

1935 

-  13    26-62 

-  13    26-52 

-11     41-58 

1-66999, 

-1     47-543 

1940 

-15    21-87 

-15    21-75 

-13     21-80 

1  -727  98n 

-2      2-908 

1945 

-17     17-13 

-17     16-97 

-15      2-02 

l-77912n 

-2     18-273 

1950 

-  19     12-40 

-  19     12-20 

-16    42-23 

l-82487n 

-2     33-640 

The  only  constants  usually  necessary  are  £0,  h  sin  0,  arid  m  =  £0  +  -• 
There  are  two  cases  between  which  the  choice  depends  on  the  nearness 
of  the  star  to  the  pole,  the  length  of  time  over  which  the  reduction 
extends,  and  the  degree  of  numerical  precision  required.  If  the 
quotient  by  dividing  the  interval  of  reduction  in  years  by  the  north 
polar  distance  in  degrees  does  not  exceed  30,  we  may,  practically, 
nearly  always  use  method  A  below,  which  requires  the  approximate 


APPENDIX    IV.  417 

value  of  p  called  pQ  in  §140,  a  correction  being  applied  by  means  of 
the  Tables  X.,  XVI,  and  XVII.  When  using  this  method,  Tables 
XII.  and  XIII.  are  not  necessary.  Within  30°  of  the  pole  we  may 
consider  the  thousandths  of  a  second  as  unimportant,  unless  an  un- 
usual degree  of  theoretical  precision  is  required. 


A.—  The  Usual  Method. 

Putting  oc0  and  8Q  for  the  given  K.A.  and  Dec.  for  epoch  T0  we 

comPute  «  =  <*„  +  £„. 

If  oc0  is  reduced  to  arc,  it  will  suffice  to  express  oc0  and  £0  to  hundredths 
of  a  minute,  reducing  the  seconds  of  f0  to  minutes.  If  we  have  a 
table  of  logarithms,  (5-place)  with  argument  in  time,  we  may  use, 
without  important  error, 


Form  the  logarithm  of 

ps  =  h  sin  6  tan  80. 

Enter  Table  XIV.  with  Arg.  log^>s  cos  a  and  take  out  log  K,  which  has 
the  same  algebraic  sign  as  p,  cos  a,  and  is  to  be  taken  from  the 
column  +  or  -  according  to  this  sign. 

Compute  \a  =  Kps  sin  a. 

Take  A^  from  Table  XV.  with  the  elapsed  time  in  years,  =  T^,  and 
a  as  the  arguments.  If  Ty  is  not  found  in  the  table,  note  that  for  any 
a,  AjO.  is  proportional  to  its  square  and  may  be  found  by  multiplying 
the  value  of  A^  for  Ty=  100  Y  by  T2. 

Take  the  factor  F  from  Table  XVI.,  and  form 


These  numbers  are  so  small  that  this  product  may  be  formed 
mentally  at  sight.  The  third  decimal  of  F  is  practically  unnecessary 
when  .F>0'100,  and  may  nearly  always  be  dropped. 

Take  the  reduction  R  from  tangent  to  arc  from  Table  XIII.  with 
argument  A0&  +  AJ&  +  A2a  =  Ata,  instead  of  which  we  may  nearly 
always  use  A0a,  without  important  error.  When  the  argument  falls 
in  the  first  column,  the  value  of  R  is  the  same  for  all  intermediate 
values  of  the  argument  before  the  next  one  following. 

Then  Aa  =  \a  +  Ajft  +  A2a  -  R, 

noting  that  R  is  always  numerically  subtractive. 

N.S.A.  2D 


418  APPENDIX   IV. 

Should  the  value  of  Aa  exceed   that  to  which  the  table  extends, 
we  subtract  log  h  from  log  A,a,  which  will  give  the  tangent  of  A&. 

tan  Aa  =  [5 -861  666]  AA 

With  this  tangent  Aa  ma}7  be  taken  from  an  ordinary  table  of  log. 
tangents.     Then 

a  =  oc0  +  Aft  +  m  =  a  +  z ••  +  Aa, 
and  if  Aa  <  6  °,  8  =  80  +  6  cos  (a  +  J  Aa)  sec  J  Aa. 


B. — Rigorous  Method. 

If  Ty  +  N.P.D.°>30,  Tables  XIV.  to  XVII.  cannot  always  be  used. 
In  this  case  Method  A  will  suffice  for  a  reduction  which  shall  be 
accurate  to  ±0*001  s.  only  when  Ty  sin  oc  tan  8  <  40,  and  will  fail  when 
Ty  sin  a  tan  8  >  40.  When,  from  this  cause,  Method  A  is  not  applicable, 
we  modify  it  as  follows  : 

Compute  a  =  oc0  +  f0, 

p  =  sin  0(tan  80  +  tan  J0  cos  a). 

Log  tan  \Q  may  be  taken  from  Table  XII.  or   computed.     As  the 
usual  tables  of  addition  and  subtraction  logarithms  are  not  convenient 
for  this  computation,  we  give  Table  XIII.     We  form 
Diff.  =  log  tan  S0  -  log  tan  \B  cos  a. 

With  this  argument,  take  from  Table  XIII.  a  logarithm  from  the 
column  Add.  to  add  to  log  tan  80  when  tan  80  and  tan  J#  cos  oc  are  of 
the  same  sign,  or  from  the  column  Subt.  to  subtract  when  they  are 
of  opposite  signs. 

We  then  have  tan  Aa  =  1  P  Sm  a    =  Kp  sin  a, 

1  -  p  cos  a 

from  which  we  compute  a  and  oc  as  in  Method  A. 

When  log^cos&<9-26,  and  5-place  logarithms  are  sufficiently  exact, 
which  will  generally  be  the  case,  we  may  use  Table  XIV.  to  find 
log(l  -p  cos  a),  entering  it  with  log^?,  cos  a  =  [4'1 38  33] p  cos  a  as  the 
argument.  If  p  cos  a  is  positive,  we  have 

log  tan  Aa  =  logp  sin  a  +  log  K+ 
if  negative,  log  tan  Ao,  =  log^>  sin  a-logK-. 

The  declination  may  be  reduced  as  in  Method  A,  or  by  the  rigorous 
formula  (19),  §138. 


APPENDIX   IV. 


419 


log  tan  \Q.     Arg.  :• 


TABLE   XII. 

-Number  of  Years 


which  the  Reduction  extends. 


The  tangent  is  positive  for  a  reduction  from  an  earlier  to  a  later  epoch, 
and  negative  in  the  opposite  case. 


Years. 

log  tan  £0. 

Years. 

log  tan  §0. 

Years. 

log  tan  iff. 

Years. 

log  tan  £0. 

Years. 

log  tan  t#. 

20 

6-9877 

45 

7-3398 

70 

7-5317 

95 

7-6643 

120 

7-7658 

25 

7-0845 

50 

7-3856 

75 

7-5617 

100 

7-6866 

125 

7-7835 

30 

7-1636 

55 

7-4270 

80 

7-5898 

105 

7-7078 

130 

7-8006 

35 

7-2306 

60 

7-4647 

85 

7-6161 

110 

7-7280 

135 

7-8170 

40 

7-2887 

65 

7-4995 

90 

7-6408 

115 

7-7473 

140 

7-8330 

Addition  and  Subtraction  Logarithms  to  the  Sixth  Decimal  Place. 
TABLE   XIII. 


Diff. 

A.orS. 

Diff. 

A.  or  8. 

Diff. 

Add. 

Subt. 

Diff. 

Add. 

Subt. 

.AAA 

.AAA 

2-20 

0-002732,, 

2749ft, 

2-60 

0-001090 

1092Qf; 

3-00 

UUU 
434m 

3-40 

"/UU 

1734 

•21 

2670? 

2686'' 

•61 

1065* 

1067f 

•01 

424 

•41 

1694 

•22 

2609 

2625 

•62 

10< 

1043 

•02 

415o 

•42 

1654 

•23 

2550 

256560 

•63 

ion2! 

101  924 

•03 

40510 

•43 

161* 

•24 

2492;!* 

2WW" 

•64 

0994^ 

0996- 

•04 

3969 

•44 

158^ 

2-25 

0-002435 

2449Kfi 

2-65 

0-00097  lw 

0973>>7 

3-05 

387Q 

3-45 

154., 

•26 

2380" 

239356 

•66 

0949:! 

095C 

•06 

378" 

•46 

151* 

•27 

2326"' 

223954 

•67 

0928 

0929 

•07 

370' 

•47 

1474 

•28 

2273* 

2286" 

•68 

0906!! 

09082! 

•08 

361. 

•48 

144^ 

•29 

2222- 

2233- 

•69 

088620° 

0888* 

•09 

3538 

•49 

i«: 

2-30 

0-0021  7  L0 

2182 

2'70 

0-000866^ 

0867n' 

3-10 

345a 

3-50 

137, 

•31 

2122" 

213250 

•71 

084620 

0848 

•11 

337' 

•51 

134* 

•32 

207448 

208448 

•72 

0827; 

08282' 

•12 

329' 

•52 

131 

•33 

2027' 

203648 

•73 

0808 

0809' 

•13 

322 

•53 

128, 

•34 

1981* 

199046 

•74 

0790;* 

0791J8 

•14 

3158 

•54 

125J 

2-35 

0-001936,, 

1944 

2-75 

0-0007721R 

077318 

3-15 

307, 

3-55 

122 

•36 

1892** 

190044 

•76 

0754® 

0755' 

•16 

300 

•60 

I09S 

•37 

1849 

185743 

•77 

0737, 

0738 

•17 

294 

•70 

087S 

•38 

180742 

18144lJ 

•78 

0720" 

0721' 

•18 

28777 

•80 

069' 

•39 

"«C 

1773:; 

•79 

0704^ 

0705- 

•19 

28°6 

•90 

o< 

2-40 

0-001726... 

1732 

2-80 

0-000688 

0689 

3-20 

274ft 

4-00 

043Q 

•41 

1686w 

1693* 

•81 

0672;;; 

0673;; 

•21 

268' 

•10 

034! 

•42 

1648* 

165439 

•82 

0657, 

0658 

•22 

262' 

•20 

°275 

•43 

161C 

1617s7 

•83 

064215 

0643" 

•23 

2566fi 

•30 

0225 

•44 

«< 

158(5 

•84 

0627;; 

0628JJ 

•24 

250^ 

•40 

°173 

2-45 

0-001538  w 

1544 

2-85 

0-000613, 

0614. 

3-25 

244, 

4-50 

014, 

•46 

1603* 

1508 

•86 

0599 

06004 

•26 

239 

•60 

oir 

•47 

1469;! 

147434 

•87 

0585; 

05864 

•27 

233 

•70 

0092 

*48 

143< 

1440;4 

•88 

0572! 

0573, 

•28 

2285 

•80 

0072 

•49 

14< 

14< 

•89 

0559J; 

0560^ 

•29 

223 

5 

•90 

0052 

2-50 

0-001371 

1376w 

2-90 

0-000546, 

°5471Q 

3-30 

218 

5-00 

004 

•51 

1340 

134432 

•91 

053412 

05352 

•31 

21  35 

•10 

003' 

•52 

in<C 

131430 

•92 

052212 

0522' 

•32 

208 

•20 

003° 

•53 

128030 

128430 

•93 

0510 

0511" 

•33 

203' 

•30 

002' 

•54 

«c 

1254~ 

•94 

0498;2 

o499;2 

•34 

1984 

•40 

002j 

2-55 

0-001222^ 

1226 

2'95 

0-000487n 

04880 

3-35 

194A 

5-50 

001 

•56 

119C 

1198" 

•96 

047611 

04762 

•36 

1904 

•60 

001 

•57 

1167* 

11702! 

•97 

0465;" 

0466° 

•37 

1855 

•70 

001° 

•58 

114126 

114426 

•98 

045510 

0455" 

•38 

1814 

•80 

001° 

•59 

1115- 

iii8re 

•99 

0444j; 

0445- 

•39 

17744 

•90 

OOlJ 

2-60 

0-001090 

1092 

3-00 

0-000434 

0435 

3-40 

173 

6-00 

000 

1 

420 


APPENDIX  IV. 


TABLE  XIV. 

Argument  =  log^4  cos  a  •   K= 


1  -p  cos  a 


Arg. 

logK. 
Arg.  + 

log  K. 
Arg.  - 

Arg. 

log*. 
Arg.  + 

logK. 
Arg.  - 

Arg. 

tog  jr. 

Arg.  + 

log  A'. 
Arg.  - 

9-20 

o-ooooi 

o-ooooi 

1-40 

0-00079 

0-00079 

1-80 

0-00200 

O-OO  199 

•30 

001 

001 

•41 

081 

081 

•81 

204 

204 

•40 

001 

001 

•42 

083 

083 

•82 

209 

208 

•50 

001 

001 

•43 

085 

085 

•83 

214 

213 

•60 

001 

001 

•44 

087 

087 

•84 

219 

218 

9  70 

0-00002 

0-00002 

1-45 

0-00089 

0-00089 

1-85 

0-00224 

0-00223 

•80 

002 

002 

•46 

091 

091 

•86 

229 

228 

•90 

003 

003 

•47 

093 

093 

•87 

235 

234 

o-oo 

003 

003 

•48 

096 

095 

•88 

240 

239 

•10 

004 

004 

•49 

098 

098 

•89 

246 

244 

0-20 

0-00005 

0-00005 

1-50 

o-oo  100 

o-ooioo 

1-90 

0-00252 

0-00250 

•30 

006 

006 

•51 

102 

102 

•91 

257 

256 

•40 

008 

008 

•52 

105 

104 

•92 

264 

262 

•50 

010 

010 

•53 

107 

107 

•93 

270 

268 

•60 

013 

013 

•54 

110 

109 

•94 

276 

274 

0-70 

0-00016 

0-00016 

1-55 

0-00112 

0-00112 

1-95 

0-00282 

0-00281 

•80 

020 

020 

•56 

115 

115 

•96 

289 

287 

•90 

025 

025 

•57 

118 

117 

•97 

296 

294 

1-00 

032 

032 

•58 

120 

120 

•98 

303 

301 

•10 

040 

040 

•59 

123 

123 

•99 

310 

308 

1-20 

0-00050 

0-00050 

1-60 

0-00  126 

0-00126 

2-00 

0-00317 

0-00315 

•21 

051 

051 

•61 

129 

129 

•01 

324 

322 

•22 

052 

052 

•62 

132 

132 

•02 

332 

329 

•23 

054 

054 

•63 

135 

134 

•03 

340 

337 

•24 

055 

055 

•64 

138 

138 

•04 

348 

345 

1-25 

0-00056 

0-00056 

1-65 

0-00141 

0-00141 

2-05 

0-00356 

0-00353 

•26 

058 

058 

•66 

145 

144 

•06 

364 

361 

•27 

059 

059 

•67 

148 

147 

•07 

373 

369 

•28 

060 

060 

•68 

151 

151 

•08 

381 

378 

•29 

062 

062 

•69 

155 

154 

•09 

390 

387 

1  30 

0-00063 

0-00063 

1-70 

0-00159 

0-00  158 

2-10 

0-00399 

0-00396 

•31 

065 

065 

•71 

162 

162 

•11 

409 

405 

•32 

066 

066 

•72 

166 

165 

•12 

418 

414 

•33 

068 

068 

•73 

170 

169 

•13 

428 

424 

•34 

069 

069 

•74 

174 

173 

•14 

438 

434 

1-35 

0-00071 

0-00071 

1-75 

0-00178 

0-00177 

2-15 

0-00448 

0-00444 

•36 

072 

072 

•76 

182 

181 

•16 

459 

454 

•37 

074 

074 

•77 

186 

186 

•17 

470 

465 

•38 

076 

076 

•78 

191 

190 

•18 

481 

475 

•39 

078 

078 

•79 

195 

194 

•19 

492 

486 

1-40 

0-00079 

0-00079 

1-80 

0-00200 

0-00199 

2-20 

0-00  503 

0-00498 

APPENDIX  IV. 


421 


TABLE   XIV.— Continued. 
Argument  =  log  ps  cos  a ;   K = 


_ 

1  -  p  cos  a 


Arg. 

log  K. 
Arg.  + 

log*. 

Arg.  - 

Arg. 

log*. 
Arg.  + 

logK. 
Arg.  - 

Arg. 

log*. 
Arg.  + 

log/r. 

Arg.  - 

2-20 

0-00503 

0-00498 

2-60 

0-01276 

0-01  240 

3-000 

0-03279 

0-03049 

•21 

515 

509 

•61 

306 

268 

•001 

287 

056 

•22 

527 

521 

•62 

337 

297 

•002 

295 

062 

•23 

540 

533 

•63 

369 

327 

•003 

303 

069 

•24 

552 

545 

•64 

401 

357 

•004 

311 

076 

2-25 

0-00565 

0-00558 

2-65 

0-01434 

0-01388 

3-005 

0-03318 

0-03083 

•26 

579 

571 

•66 

468 

420 

•006 

326 

090 

•27 

592 

584 

•67 

503 

453 

•007 

334 

097 

•28 

606 

598 

•68 

539 

486 

•008 

342 

103 

•29 

620 

612 

•69 

575 

520 

•009 

350 

110 

2-30 

0-00635 

0-00626 

2-70 

0-01612 

0-01  555 

3-010 

0-03358 

0-03117 

•31 

650 

640 

•71 

651 

590 

•on 

366 

124 

•32 

665 

655 

•72 

690 

627 

•012 

375 

131 

•33 

681 

670 

•73 

730 

664 

•013 

383 

138 

•34 

696 

686 

•74 

771 

702 

•014 

391 

145 

2-35 

0-00713 

0-00701 

2-75 

0-01813 

0-01  741 

3-015 

.0-03399 

0-03152 

•36 

730 

718 

•76 

857 

780 

•016 

407 

159 

•37 

747 

734 

•77 

901 

821 

•017 

415 

166 

•38 

764 

751 

•78 

946 

862 

•018 

423 

173 

•39 

782 

768 

•79 

992 

905 

•019 

432 

180 

240 

0-00801 

0-00786 

2-80 

0-02040 

0-01948 

3-020 

0-03440 

0-03  187 

•41 

819 

804 

•81 

089 

993 

•021 

448 

194 

•42 

839 

823 

•82 

138 

0-02038 

•022 

456 

201 

•43 

858 

842 

•83 

190 

084 

•023 

465 

209 

•44 

879 

861 

•84 

242 

132 

•024 

473 

216 

2-45 

0-00899 

0-00881 

2-85 

0-02296 

0-02180 

3-025 

0-03481 

0-03223 

•46 

921 

901 

•86 

350 

230 

•026 

490 

230 

•47 

942 

922 

•87 

407 

280 

•027 

498 

237 

•48 

964 

943 

•88 

464 

332 

•028 

506 

244 

•49 

987 

965 

•89 

524 

385 

•029 

515 

252 

2-50 

0-01010 

0-00987 

2-90 

0-02584 

0-02439 

3-030 

0-03523 

0-03259 

•51 

034 

0-01010 

•91 

646 

494 

•031 

532 

266 

•52 

059 

033 

•92 

710 

551 

•032 

540 

273 

•53 

084 

057 

•93 

775 

608 

•033 

549 

281 

•54 

109 

082 

•94 

842 

667 

•034 

557 

288 

2-55 

0-01  135 

0-01  106 

2-95 

0-02910 

0-02727 

3-035 

0-03566 

0-03295 

•56 

162 

132 

•96 

980 

789 

•036 

574 

302 

•57 

190 

158 

•97 

0-03052 

852 

•037 

583 

310 

•58 

218 

f84 

•98 

126 

916 

•038 

592 

317 

•59 

246 

212 

•99 

202 

982 

•039 

600 

324 

2-60 

0-01276 

0-01240 

3-00 

0-03279 

0-03049 

3*040 

0-03609 

0-03332 

422 


APPENDIX   IV. 


TABLE   XIV.— Continued. 
Argument  =  \ogpscosa;   K= 


1 


1  -  p  cos  a 


Arg. 

log*. 
Arg.  + 

log  A. 
Arg.  - 

Arg. 

log  A. 
Arg.  + 

log  A'. 

Arg.  - 

Arg 

log  A. 
Arg.  + 

log  A. 

Arg.  - 

3-040 

0-03609 

0-03332 

3-080 

0-03973 

0-03640 

3-120 

0-04377 

0-03976 

•041 

618 

339 

•081 

983 

648 

•121 

387 

985 

•042 

626 

347 

•082 

993 

656 

•122     398 

993 

•043 

635 

354 

•083 

0-04002 

664 

•123 

409  0*04002 

•044 

644 

362 

•084 

012 

672 

•124 

419      Oil 

3-045 

0-03653 

0-03369 

3-085 

0-04022 

0-03681 

3-125 

0-04430 

0-04020 

•046 

661 

376 

•086 

031 

689 

•126      441 

029 

•047 

670 

384 

•087 

041 

697 

•127      452 

037 

•048 

679 

391 

•088 

051 

705 

•128      462 

046 

•049 

688 

399 

•089 

061 

713 

•129 

473 

055 

3-050 

0-03697 

0-03406 

3-090 

0-04070 

0-03721 

3-130 

0-04484 

0-04064 

•051 

706 

414 

•091 

080 

730 

•131 

495 

073 

•052 

714 

422 

•092 

090 

738 

•132 

506 

082 

•053 

723 

429 

•093 

100 

746 

•133 

517 

091 

•054 

732 

437 

•094 

110 

754 

•134 

528 

100 

3-055 

0-03741 

0-03444 

3-095 

0-04120 

0-03763 

3-135 

0-04539 

0-04  109 

•056 

750 

452 

•096 

130 

771 

•136 

550 

118 

•057 

759 

460 

•097 

140 

779 

•137 

561 

127 

•058 

768 

467 

•098 

150 

788 

•138 

572 

136 

•059 

111 

475 

•099 

160 

796 

•139 

583 

145 

3-060 

0-03787 

0-03483 

3-100 

0-04170 

0-03804 

3-140 

0-04594 

0-04  154 

•061 

796 

490 

•101 

180 

813 

•141 

605 

164 

•062 

805 

498 

•102 

190 

821 

•142 

617 

173 

•063 

814 

506 

•103 

200 

830 

•143 

628 

182 

•064 

823 

514 

•104 

210 

838 

•144 

639 

191 

3-065 

0-03832 

0-03521 

3-105 

0-04221 

0-03847 

3-145 

0*04650 

0-04200 

•066 

842 

529 

•106 

231 

855 

•146 

662 

210 

•067 

851 

537 

•107 

241 

864 

•147 

673 

219 

•068 

860 

545 

•108 

251 

872 

•148 

684 

228 

•069 

869 

553 

•109 

262 

881 

•149 

696 

237 

3-070 

0-03  879 

0-03561 

3'110 

0-04272 

0-03889 

3-150 

0-04707 

0-04247 

•071 

888 

569 

•111 

282 

898 

•151 

719 

256 

•072 

898 

576 

•112 

293 

906 

•152 

730 

265 

•073 

907 

584 

•113 

303 

915 

•153 

742 

275 

•074 

916 

592 

•114 

314 

924 

•154 

753 

284 

3-075 

0-03926 

0-03600 

3-115 

0-04324 

0-03932 

3-155 

0-04765 

0-04293 

•076 

935 

608 

•116 

335 

941 

•156 

111 

303 

•077 

945 

616 

•117 

345 

950 

•157 

788 

312 

•078 

954 

624 

•118 

356 

958 

•158 

800 

322 

•079 

964 

632 

•119 

366 

967 

•159 

812 

331 

3-080 

0-03973 

0-03640 

3-120 

0-04377 

0-03976 

3-160 

0-04823 

0-04341 

APPENDIX   IV. 


TABLE   XIV.— Continued. 
Argument  =  logj?g  cos  a ;   K= 


1  -J9COSCI 


Arg. 

log*. 
Arg.  + 

log  #. 

Arg.  - 

Arg. 

log*. 
Arg.  + 

log*. 
Arg.  - 

Arg. 

log  K. 

Arg.  + 

log  K. 
Arg.  - 

3-160 

0-04823 

0-04341 

3-200 

0-05318 

0-04737 

3-240 

0-05868 

0-05168 

•161 

835 

350 

•201 

331 

748 

•241 

882 

180 

•162 

847 

360 

•202 

344 

758 

•242 

897 

191 

•163 

859 

369 

•203 

358 

769 

•243 

911 

202 

•164 

870 

379 

•204 

371 

779 

•244 

926 

213 

3-165 

0-04882 

0-04389 

3-205 

0-05384 

0-04789 

3-245 

0-05940 

0-05225 

•166 

894 

398 

•206 

397 

800 

•246 

955 

236 

•167 

906 

408 

•207 

410 

810 

•247 

970 

247 

•168 

918 

418 

•208 

424 

821 

•248 

984 

259 

•169 

930 

427 

•209 

437 

831 

•249 

999 

270 

3-170 

0-04942 

0-04437 

3-210 

0-05450 

0-04842 

3-250 

0-06014 

0-05282 

•171 

954 

447 

•211 

464 

852 

•251 

029 

293 

•172 

966 

456 

•212 

477 

862 

•252 

044 

305 

•173 

979 

466 

•213 

491 

874 

•253 

059 

316 

•174 

991 

476 

•214 

504 

884 

•254 

074 

328 

3-175 

0-05003 

0-04486 

3-215 

0-05518 

0-04895 

3-255 

0-06089 

0-05339 

•176 

015 

496 

•216 

531 

906 

•256 

104 

351 

•177 

027 

505 

•217 

545 

916 

•257 

119 

362 

•178 

040 

515 

•218 

558 

927 

•258 

134 

374 

•179 

052 

525 

•219 

572 

938 

•259 

149 

386 

3-180 

0-05064 

0-04535 

3-220 

0-05586 

0-04949 

3-260 

0-06165 

0-05397 

•181 

077 

545 

•221 

599 

959 

•261 

180 

409 

•182 

089 

555 

•222 

613 

970 

•262 

195 

421 

•183 

102 

565 

•223 

627 

981 

•263 

211 

433 

•184 

114 

575 

•224 

641 

992 

•264 

226 

444 

3-185 

0-05127 

0-04585 

3-225 

0-05655 

0-05003 

3-265 

0-06241 

0-05456 

•186 

139 

595 

•226 

669 

014 

•266 

257 

468 

187 

152 

605 

•227 

683 

024 

•267 

272 

480 

•188 

164 

615 

•228 

697 

035 

•268 

288 

492 

•189 

177 

625 

•229 

711 

046 

•269 

304 

504 

3-190 

0-05  190 

0-04635 

3-230 

0-05725 

0-05057 

3-270 

0-06319 

0-05515 

•191 

202 

645 

•231 

739 

068 

•271 

335 

527 

•192 

215 

655 

•232 

753 

079 

•272 

351 

539 

•193 

228 

666 

•233 

767 

090 

•273 

366 

551 

•194 

241 

676 

•234 

781 

101 

•274 

382 

563 

3-195 

0-05254 

0-04686 

3-235 

0-05796 

0-05113 

3*275 

0-06398 

0-05575 

•196 

266 

696 

•236 

810 

124 

•276 

414 

587 

•197 

279 

707 

•237 

825 

135 

•277 

430 

599 

•198 

292 

717 

•238 

839 

146 

•278 

446 

612 

•199 

305 

727 

•239 

853 

157 

•279 

462 

624 

3-200 

0-05318 

0-04737 

3-240 

0-05868 

0-05168 

3-280 

0-06478 

0-05636 

424 


APPENDIX   IV. 


TABLE   XIV.— Concluded. 


Argument  =  log  ps  cos  a  \  K=  - — 


1 


Arg. 

logK. 
Arg.  + 

log  A\ 

Arg.  - 

Arg. 

log  K. 
Arg.  + 

Arf  .  - 

Arg. 

log  JT. 
Arg.  + 

logK. 
Arg.  - 

3-280 

0-06478 

0-05636 

3-320 

0-07  157 

0-06  143 

3-360 

0-07915 

0-06692 

•281 

494 

648 

•321 

175 

156 

•361 

935 

706 

•282 

510 

660 

•322 

193 

169 

•362 

955 

721 

•283 

526 

673 

•323 

211 

183 

•363 

975 

735 

•284 

543 

685 

•324 

229 

196 

•364 

995 

749 

3-285 

0-06559 

0-05697 

3-325 

0-07247 

0-06209 

3-365 

0-08015 

0-06764 

•286 

575 

709 

•326 

266 

223 

•366 

035 

778 

•287 

592 

722 

•327 

284 

236 

•367 

056 

793 

•288 

608 

734 

•328 

302 

249 

•368 

076 

807 

•289 

624 

746 

•329 

320 

263 

•369 

097 

822 

3-290 

0-06641 

0-05759 

3-330 

0-07339 

0-06276 

3-370 

0-08117 

0-06836 

•291 

658 

771 

•331 

357 

290 

•371 

138 

851 

•292 

674 

784 

•332 

376 

303 

•372 

158 

865 

•293 

691 

796 

•333 

394 

317 

•373 

179 

880 

•294 

707 

809 

•334 

413 

330 

•374 

200 

895 

3-295 

0-06724 

0-05821 

3-335 

0-07432 

0-06344 

3-375 

0-08221 

0-06909 

•296 

741 

834 

•336 

450 

357 

•376 

242 

924 

•297 

758 

846 

•337 

469 

371 

•377 

262 

939 

•298 

775 

859 

•338 

488 

385 

•378 

283             954 

•299 

792 

872 

•339 

507 

398 

•379 

304  |          969 

3-300 

0-06808 

0-05884 

3-340 

0-07.525 

0-06412 

3-380 

0-08326 

0-06983 

•301 

825 

897 

•341 

544 

426 

•381 

347 

998 

•302 

843 

910 

•342 

563 

440 

•382 

368     0-07013 

•303 

860 

922 

•343 

582 

453 

•383 

389  :          028 

•304 

877 

935 

•344 

602 

467 

•384 

411             043 

3-305 

0-06894 

0-05948 

3-345 

0-07  621 

0-06481 

3-385 

0-08432     0-07058 

•306 

911 

961 

•346 

640 

495 

•386 

453             073 

•307 

928 

974 

•347 

659 

509 

•387 

475             088 

•308 

946 

987 

•348 

679 

523 

•388 

496 

103 

•309 

963 

999 

•349 

698 

537 

•389 

518 

118 

3-310 

0-06980 

0-06012 

3-350 

0-07717 

0-06551 

3-390 

0-08540 

0-07  133 

•311 

998 

025 

•351 

737 

565 

•391 

562 

149 

•312 

0-07015 

038 

•352 

756 

579 

•392 

583 

164 

•313 

033 

051 

•353 

776 

593 

•393 

605 

179 

•314 

051 

064 

•354 

796 

607 

•394 

627 

194 

3-315 

0-07068 

0-06077 

3-355 

0-07815 

0-06621 

3-395 

0-08649 

0-07209 

•316 

086 

090 

•356 

835 

635 

•396 

671 

225 

•317 

104 

103 

•357 

855 

649 

•397 

693 

240 

•318 

122 

117 

•358 

875 

664 

•398 

716 

255 

•319 

139 

130 

•359 

895 

678 

•399 

738 

271 

3-320 

0-07  157 

0-06  143 

3-360 

0-07915 

0-06692 

3-400 

0-08760 

0-07286 

APPENDIX  IV. 


425 


TABLE  XV. 

\a.     Arg.  a,  or  a  -  1 2  ft.     Arg.  at  top,  ±  Ty. 


Arg. 

25  Y. 

30  Y. 

40  Y. 

50  Y. 

COY. 

70  Y. 

SOY. 

100  Y.     120  Y. 

145  Y. 

Arg. 

h.     m. 

3. 

s. 

s. 

s. 

s. 

s. 

s. 

s. 

s. 

s. 

ra.     h. 

0     o 

o-ooo 

o-ooo 

o-ooo 

o-ooo 

o-ooo 

o-ooo 

o-ooo 

o-ooo 

o-ooo 

o-ooo 

o  12 

10 

•002 

•002 

•004 

•007 

•010 

•014 

•018 

•028 

•040 

•059 

50 

20 

•004 

•005 

•009 

•014 

•020 

•027 

•036 

•056 

•080 

•117 

40 

30 

•005 

•008 

•013 

•021 

•030 

•041 

•054 

•083 

•119 

•174 

30 

40 

•007 

•010 

•017 

•027 

•039 

•054 

•072 

•109 

•157 

•230 

20 

50 

•008 

•012 

•022 

•034 

•048 

•067 

•088 

•135 

•194 

284 

10 

1     o 

o-oio 

0-014 

0-026 

0-040 

0-057 

0-079 

0-104 

0-160 

0-230 

0-336 

o  11 

10 

•012 

•016 

•029 

•046 

•066 

•091 

•120 

•184 

•264 

•386 

50 

20 

•013 

•018 

•033 

•051 

•074 

•102 

•134 

•206 

•296 

•433 

40 

30 

•014 

•020 

•036 

•057 

•081 

•112 

•148 

•226 

•325 

•476 

30 

40 

•015 

•022 

•039 

•062 

•088 

•121 

•160 

•245 

•352 

•516 

20 

50 

•016 

•024 

•042 

•066 

•094 

•129 

•171 

•262 

•377 

•551 

10 

2       0 

0-017 

0-025 

0-044 

0-069 

o-ioo 

0-137 

0-181 

0-277 

0-398 

0-583 

o  10 

10 

•018 

•026 

•046 

•072 

•104 

•143 

•189 

•290 

•417 

•610 

50 

20 

•019 

•027 

•048 

•075 

•108 

•148 

•196 

•301 

•432 

•632 

40 

30 

•019 

•028 

•049 

•077 

•111 

•153 

•202 

•309 

•444 

•650 

30 

40 

•020 

•029 

•050 

•079 

•113 

•156 

•206 

•315 

•453 

•663 

20 

50 

•020 

•029 

•051 

•080 

•115 

•157 

•208 

•319 

•458 

•670 

10 

3     o 

0-020 

0-029 

0-051 

0-080 

0-115 

0-158 

0-209 

0-320 

0-460 

0-673 

0     9 

10 

•020 

•029 

•051 

•080 

•115 

•157 

•208 

•319 

•458 

•670 

50 

20 

•020 

•029 

•050 

•079 

•113 

•156 

•206 

•315 

•453 

•663 

40 

30 

•019 

•028 

•049 

•077 

•111 

•153 

•202 

•309 

•444 

•650 

30 

40 

•019 

•027 

•048 

•075 

•108 

•148 

•196 

•301 

•432 

•632 

20 

50 

•018 

•026 

•046 

•072 

•104 

•143 

•189 

•290 

•417 

•610 

10 

4     o 

0-017 

0-025 

0044 

0-069 

o-ioo 

0-137 

0-181 

0-277 

0-398 

0-583 

o    8 

10 

•016 

•024 

•042 

•066 

•094 

•129 

•171 

•262 

•377 

•551 

50 

20 

•015 

•022 

•039 

•062 

•088 

•121 

•160 

•245 

•352 

•516 

40 

30 

•014 

•020 

036 

•057 

•081 

•112 

•148 

•226 

•325 

•476 

30 

40 

•013 

•018 

•033 

•051 

•074 

•102 

•134 

•206 

•296 

•433 

20 

50 

•012 

•016 

•029 

•046 

•066 

•091 

•120 

•184 

•264 

•386 

10 

5     o 

o-oio 

0-014 

0-026 

0-040 

0-057 

0-079 

0-104 

0-160 

0-230 

0-336 

o    7 

10 

•008 

•012 

•022 

•034 

•048 

•067 

•088 

•135 

•194 

•284 

50 

20 

•007 

•010 

•017 

•027 

•039 

•054 

•072 

•109 

•157 

•230 

40 

30 

•005 

•008 

•013 

•021 

•030 

•041 

•054 

•083 

•119 

•174 

30 

40 

•004 

•005 

•009 

•014 

•020 

•027 

•036 

•056 

•080 

•117 

20 

50 

•002 

•002 

•004 

•007 

•010 

•014 

•018 

•028 

•040 

•059 

10 

6     o 

o-ooo 

o-ooo 

o-ooo 

o-ooo 

o-ooo 

o-ooo 

o-ooo 

o-ooo 

o-ooo 

o-ooo 

o    6 

The  correction  Axa  is  positive  when  the  argument  is  on  the  left,  and  negative  when 
on  the  right.     The  sign  is  the  same  whether  Ty  is  positive  or  negative. 


426 


APPENDIX   IV. 


TABLE    XVI. 

To  form  A2a  =  Aj«  x  F.     Arg.  log  pt  cos  a. 


Arg. 

F. 

Arg. 

F. 

Arg. 

F. 

Arg.  + 

Arg.  - 

Arg.  + 

Arg.  - 

Arg.  + 

Arg.  - 

1-80 

+  •009 

-•009 

3-00 

+  •163 

-•131 

3-20 

+  "278 

-  -196 

1-90 

•012 

•012 

•01 

•167 

•133 

•21 

•285 

•200 

2-00 

•015 

•015 

•02 

•172 

•136 

•22    I       '293 

•204 

•10 

•019 

•018 

•03 

•176 

•139 

•23           '302 

•208 

•20 

•023    j      '023 

•04 

•181 

•142 

•24 

•310 

•212 

2-30 

+  -030 

-•029 

3-05 

+  •186 

-•145 

3-25 

+  '319 

-•216 

•40 

•038          -036 

•06 

•191 

•148 

•26 

•328 

•220 

•50 

•048          '045 

•07 

•196 

•151 

•27 

•337 

•224 

•60 

•060 

•055 

•08 

•201 

•154 

•28 

•347 

•228 

•70 

•077    i      -069 

•09 

•206 

•158 

•29 

•357 

•232 

2'80 

+  -098       -  '086 

3-10 

+  -212 

-•161 

3-30 

+  -368 

-•237 

•83 

•106          -091 

•11 

•217 

•164 

•31 

•379 

•241 

•86 

•115          '097 

•12 

•223 

•168 

•32 

•890 

"246 

•89 

•124          -104 

•13 

•229 

•171 

•33 

•402 

•250 

•92 

•133          -110 

•14 

•235 

•175 

•34 

•414 

•255 

•95 

+  •143       -'118 

3-15 

+  -242    !  -  -178 

3-35 

+  •427 

-  -260 

•96 

•147           -120 

•16 

•249          -182 

•36 

•440  . 

•265 

•97 

•151           '123 

•17 

•256          '185 

•37 

•454 

•271 

•98 

•155          '126 

•18 

•263          -189 

•38 

•468 

•278 

•99 

•159          -128 

•19 

•270          -192 

•39 

•482 

•285 

300       +'163 

-•131 

3-20 

+  "278 

-•196 

3-40 

+  •496 

-•292 

TABLE   XVII. 
Red.  from  tan.  to  arc.     Arg.  =  A0«  +  Ajfl  +  A2a  = 


A,a. 

R. 

A,a 

R. 

A,a. 

R. 

A,a. 

R. 

Ata. 

R. 

8. 

s. 

s. 

s. 

s. 

3. 

s. 

s. 

s. 

s. 

76 

•001 

200 

•014 

400 

•113 

600 

•381 

800 

•901 

96 

•002 

210 

•016 

410 

•121 

610 

•402 

810 

•936 

112 

•003 

220 

•019 

420 

•129 

620 

•423 

820 

•971 

126 

•004 

230 

•022 

430 

•139 

630 

•443 

830 

1-007 

136 

•005 

240 

•024 

440 

•150 

640 

•464 

840 

1-044 

146 

•006 

250 

•028 

450 

•160 

650 

•485 

850 

1-081 

155 

•007 

260 

•031 

460 

•171 

660 

•509 

860 

1-119 

162 

•008 

270 

•035 

470 

•182 

670 

•533 

870 

1-158 

169 

•009 

280 

•039 

480 

•194 

680 

•556 

880 

1-198 

175 

•010 

290 

•043 

490 

•207 

690 

•580 

890 

1-240 

182 

•Oil 

300 

•048 

500 

•220 

700 

•604 

900 

1-282 

187 

•012 

310 

•052 

510 

•233 

710 

•632 

910 

1-326 

192 

•013 

320 

•057 

520 

•247 

720 

•660 

920 

1-370 

197 

•014 

330 

•063 

530 

•262 

730 

•687 

930 

T415 

202 

•015 

340 

•069 

540 

•277 

740 

•715 

940 

1-461 

206 

•016 

350 

•075 

550 

•293 

750 

•743 

950 

1-507 

210 

•017 

360 

•082 

560 

•311 

760 

•774 

960 

1-555 

215 

•018 

370 

•090 

570 

•329 

770 

805 

970 

1-602 

219 

•019 

380 

•097 

580 

•346 

780 

•837 

980 

1-652 

223 

•020 

390 

•105 

590 

•364 

790 

•869 

990 

1-704 

227 

•021 

400 

•113 

600 

•381 

800 

•901 

1000 

1-757 

— Enter  first  column  with  number  equal  to  or  next  smaller  than  the 
given  argument. 


APPENDIX   IV. 


427 


TABLE    XVIL— Concluded. 
Red.  from  tan.  to  arc.     Arg.  =  AQa  +  A^  +  A2& 


A,«. 

R. 

.,-,. 

R. 

A,«, 

R. 

v, 

R, 

A,.. 

R. 

1000 

1-76 

1250 

343 

1500 

5-91 

1750 

s. 
9-36 

S. 

2000 

13^92 

1005 

1-79 

1255 

3-47 

1505 

5-96 

1755 

9-44 

2005 

14-03 

1010 

1-81 

1260 

3-51 

1510 

6-02 

1760 

9-52 

2010 

14-13 

1015 

•84 

1265 

3-55 

1515 

6-08 

1765 

9-60 

2015 

14-24 

1020 

•87 

1270 

3-59 

1520 

6-14 

1770 

9-68 

2020 

14-34 

1025 

•89 

1275 

3-63 

1525 

6-20 

1775 

9-76 

2025 

14-45 

1030 

•92 

1280 

3-68 

1530 

6-26 

1780 

9-84 

2030 

14-56 

1035 

•95 

1285 

3-72 

1535 

6-32 

1785 

9-92 

2035 

14-66 

1040 

•98 

1290 

3-76 

1510 

6-38 

1790 

10-01 

2040 

14-77 

1045 

2-01 

1295 

3-81 

1545 

6-44 

1795 

10-09 

2045 

14-88 

1050 

2-03 

1300 

3-85 

1550 

6-51 

1800 

10-18 

2050 

14-99 

1055 

2-06 

1305 

3-90 

1555 

6-57 

1805 

10-26 

2055 

15-10 

1060 

2-09 

1310 

3-94 

1560 

6-64 

1810 

10-35 

2060 

15-21 

1065 

2-12 

1315 

3-99 

1565 

6-70 

1815 

10-43 

2065 

15-32 

1070 

2-15 

1320 

4-03 

1570 

6-77 

1820 

10-52 

2070 

15-43 

1075 

2-18 

1325 

4-08 

1575 

6-83 

1825 

10-60 

2075 

15-54 

1080 

2-21 

1330 

4-12 

1580 

6-89 

1830 

10-69 

2080 

15-65 

1085 

2-24 

1335 

4-17 

1585 

6-96 

1835 

10-77 

2085 

15-76 

1090 

2-28 

1340 

4-22 

1590 

7-03 

1840 

10-86 

2090 

15-87 

1095 

2-31 

1345 

4-26 

1595 

7-10 

1845 

10-95 

2095 

15-99 

1100 

2-34 

1350 

4-31 

1600 

7-17 

1850 

11-04 

2100 

16-10 

1105 

2-37 

1355 

4-36 

1605 

7-23 

1855 

11-13 

2105 

16-22 

1110 

2-40 

1360 

4-41 

1610 

7-29 

1860 

11-22 

2110 

16-33 

1115 

2-43 

1365 

4-45 

1615 

7-36 

1865 

11-31 

2115 

16-45 

1120 

2-47 

1370 

4-50 

1620 

7-43 

1870 

11-40 

2120 

16-56 

1125 

2-50 

1375 

4-55 

1625 

7-50 

1875 

11-49 

2125 

16-68 

1130 

2-53 

1380 

4-60 

1630 

7-57 

1880 

11-58 

2130 

16-79 

1135 

2-56 

1385 

4-65 

1635 

7-64 

1885 

11-68 

2135 

16-91 

1140 

2-60 

1390 

4-70 

1640 

7-71 

1890 

11-77 

2140 

17-03 

1145 

2-63 

1395 

4-75 

1645 

778 

1895 

11-86 

2145 

17-15 

1150 

2-67 

1400 

4-81 

1650 

7-85 

1900 

11-96 

2150 

17-27 

1155 

2-70 

1405 

4-86 

1655 

7-92 

1905 

12-05 

2155 

17-39 

1160 

2-74 

1410 

4-91 

1660 

7-99 

1910 

12-14 

2160 

17-51 

1165 

2-77 

1415 

4-96 

1665 

8-06 

1915 

12-24 

2165 

17-63 

1170 

2-81 

1420 

5-01 

1670 

8-14 

1920 

12-33 

2170 

1775 

1175 

2-85 

1425 

5-07 

1675 

8-21 

1925 

12-43 

2175 

17-87 

1180 

2-88 

1430 

5-12 

1680 

8-28 

1930 

12-53 

2180 

17-99 

1185 

2-92 

1435 

5-17 

1685 

8-35 

1935 

12-62 

2185 

18-12 

1190 

2-96 

1440 

5-23 

1690 

8-43 

1940 

12-72 

2190 

18-24 

1195 

2-99 

1445 

5-28 

1695 

8-50 

1945 

12-82 

2195 

18-36 

1200 

3-03 

1450 

5-34 

1700 

8-58 

1950 

12-92 

2200 

18-49 

1205 

3-07 

1455 

5-39 

1705 

8-65 

1955 

13-02 

2205 

18-61 

1210 

3-11 

1460 

5-45 

1710 

8-73 

1960 

13-11 

2210 

18-74 

1215 

3-15 

1465 

5-51 

1715 

8-81 

1965 

13-21 

2215 

18-86 

1220 

3-19 

1470 

5-56 

1720 

8-89 

1970 

13-31 

2220 

18-99 

1225 

3-22 

1475 

5-62 

1725 

8-96 

1975 

13-42 

2225 

19-12 

1230 

3-26 

1480 

5-67 

1730 

9-04 

1980 

13-52 

2230 

19-25 

1235 

3-30 

1485 

5-73 

1735 

9-12 

1985 

13-62 

2235 

19-38 

1240 

3-34 

1490 

5-79 

1740 

9-20 

1990 

13-72 

2240 

19-50 

1245 

3-39 

1495 

5-85 

1745 

9-28 

1995 

13-82 

2245 

19-63 

1250 

3-43 

1500 

5-91 

1750 

9-36 

2000 

13-92 

2250 

19-76 

APPENDIX   V. 


Reduction  of  Struve-Peters  Centennial  Precessions  to  the  Adopted  Values. 


=  -  0-035  s.  sin  a  tan  5  -  0'038  s. 
=  -0" '53  cos  a. 


TABLE  XVIII. 


Arg. 
RA. 

-35  sin  a. 

-53  cos  a. 

h.  m. 
0   0 

-  0- 

-53  + 

m.  b. 
0  12 

10 

-  2- 

-53 

50 

20 

—  3  — 

-53 

40 

30 

-  5- 

-52 

30 

40 

-  6- 

-52 

20 

50 

-  8- 

-52 

10 

1   0 

—  9  — 

-51  + 

0  11 

10 

-11- 

-50 

50 

20 

-12- 

-50 

40 

30 

-13- 

-49 

30 

40 

-15- 

-48 

20 

50 

-16- 

-46 

10 

2   0 

-18- 

-45  + 

0  10 

10 

-19- 

-44 

50 

20 

-20- 

-43 

40 

30 

-21- 

-42 

30 

40 

-22- 

-41 

20 

50 

-24- 

-39 

10 

3   0 

-25- 

-37  + 

0   9 

10 

-26- 

-36 

50 

20 

-27- 

-34 

40 

30 

-28- 

-32 

30 

40 

-29- 

-30 

20 

50 

-30- 

-28 

10 

4   0 

-30- 

-26  + 

0   8 

10 

-31- 

-24 

50 

20 

-32- 

-22 

40 

30 

-32- 

-20 

30 

40 

-33- 

-18 

20 

50 

-33- 

-16 

10 

5   0 

-34- 

-14  + 

0   7 

10 

-34- 

-11 

50 

20 

-34- 

_  g 

40 

30 

-35- 

_  7 

30 

40 

-35- 

—  5 

20 

50 

-35- 

_  2 

10 

6   0 

-35- 

-  0  + 

0   6 

Arg. 

The  algebraic  sign  is  on  the  same 
side  of  the  number  as  the  argument. 

When  the  R.  A.  exceeds  12  h.  enter 
with  Arg.  R.  A.  - 12  h.  and  change 
the  sign. 


TABLE  XIX. 


S. 

Nat. 
tan  8. 

8. 

Nat. 
tanS. 

0 

0 

•000 

o 
45 

1-000 

1 

•017 

46 

1-036 

2 

•035 

47 

1-072 

3 

•052 

48 

•111 

4 

•070 

49 

•150 

5 

•087 

50 

•192 

6 

•105 

51 

•235 

7 

•123 

52 

•280 

8 

•141 

53 

•327 

9 

•158 

54 

•376 

10 

•176 

55 

•428 

11 

•194 

56 

•483 

12 

•213 

57 

•540 

13 

•231 

58 

•600 

14 

•249 

59 

•664 

15 

•268 

60 

•732 

16 

•287 

61 

•804 

17 

•306 

62 

•881 

18 

•325 

63 

1-963 

19 

•344 

64 

2-050 

20 

•364 

65 

2-145 

21 

•384 

66 

2-246 

22 

•404 

67 

2-356 

23 

•424 

68 

2-475 

24 

•445 

69 

2-605 

25 

•466 

70 

2-747 

26 

•488 

71 

2-904 

27 

•510 

72 

3-078 

28 

•532 

73 

3-271 

29 

•554 

74 

3-487 

30 

•577 

75 

3-732 

31 

•601 

76 

4-011 

32 

•625 

77 

4-331 

33 

•649 

78 

4-705 

34 

•675 

79 

5-145 

35 

•700 

80 

5-671 

36 

•727 

81 

6-314 

37 

•754 

82 

7-115 

38 

•781 

83 

8-144 

39 

•810 

84 

9-514 

40 

•839 

85 

11-430 

41 

•869 

86 

14-301 

42 

•900 

87 

19-081 

43 

•933 

88 

28-636 

44 

•966 

89 

57-290 

45 

1-000 

90 

00 

APPENDIX   VI. 

CONVERSION    OF   LONGITUDE   AND    LATITUDE   INTO 

RIGHT  ASCENSION  AND  DECLINATION, 

AND    VICE   VERSA. 

e  =  23°  27'  0". 

Precepts  for  the  use  of  Table  XX.  : 

To   Convert   Longitude   and  Latitude   into   Right   Ascension   and 
Declination  : 

A  or  A  -180°  =  *, 

tan^?  =  a  tan  (j3  +  B\ 
tan  8  =  b  tan  (/3  +  B)  cos  p 


«.  =  h  +  A  -p. 
Rules  for  Algebraic  Sign. 

Sign  of  a  is  that  of  cos  A, 
„  B  „  sin  A, 
„  A  „  tan  A, 
„  b  always  +  . 

To  Convert  Right  Ascension  and  Declination  into  Longitude  and 
Latitude  : 

ex.  or  a-  180°  =  &, 

tan  q  =  a  tan  (8  —  B), 
tan  /5  =  b  tan  (8  -  B)  cos  q 
or  =  b  sin  (8  -  B), 


Rules  for  Algebraic  Sign. 

Sign  of  a  is  that  of  cos  oc, 

,,     B          „      sin  a, 

,,     A  „       tana, 

„      b  always  +. 
The  following  approximate  formula  may  be  used  when  /3  <  10°  : 


P  =  b(8-B). 
sec  /3  may  be  put=  1  when  /3  <  4°, 

In  using  Table  XXI.  the  algebraic  sign  of  the  coefficient  is  that  on 
the  same  side  as  the  argument. 


430  APPENDIX   VI. 

TABLE   XX. 

Conversion  of  Longitude  and  Latitude  into  Eight  Ascension  and 
Declination,  and  vice  versa. 

e  =  23°  27'  0". 


ft. 

k. 

A. 

log  «. 

a. 

B. 

Diff. 

log  b. 

b. 

k. 

k. 

o 

h.  m. 

0 

, 

h.  m. 

0 

0 

0  0 

o  o-o 

9-5998 

0-3980 

o  o-o 

fta.c\ 

9-9626 

0-9174 

12   0 

180 

1 

0  4 

0  5-4 

9-5998 

0-3979 

0  26-0 

/O  U 

9«  -A 

9-9626 

0-9174 

11  56 

179 

2 

0  8 

0  10-8 

9-5996 

0-3977 

0  52-0 

±n  \) 

Oft  -A 

9-9626 

0-9175 

11  52 

178 

3 

0  12 

0  16-2 

9-5992 

0-3974 

1  18-0 

wD  U 
9ft  .A 

9-9627 

0-9176 

11  48 

177 

4 

0  16 

0  21-5 

9-5988 

0-3970 

1  44-0 

-iO  U 

25-9 

9-9628 

0-9178 

11  44 

176 

5 

0  20 

0  26-9 

9-5982 

0-3964 

2  9-9 

ox  .Q 

9-9629 

0-9181 

11  40 

175 

g 

0  24 

0  32-1 

9-5974 

0-3958 

2  35-8 

zo  y 

9X.t> 

9-9630 

0-9184 

11  36 

174 

7 

0  28 

0  37-4 

9-5966 

0-3950 

3  1-6 

<£O  O 

O**  ." 

9-9632 

0-9187 

11  32 

173 

8 

0  32 

0  42-6 

9-5956 

0-3941 

3  27-3 

2o  / 

9X.fi 

9-9634 

0-9191 

11  28 

172 

9 

0  36 

0  47-7 

9-5944 

0-3931 

3  52-9 

ZO  O 

25-6 

9-9636 

0-9195 

11  24 

171 

10 

0  40 

0  52-8 

9-5932 

0-3919 

4  18-5 

rtC  .A 

9-9638 

0-9200 

11  20 

170 

11 

0  44 

0  57-8 

9-5918 

0-3906 

4  43-9 

Z!)  4 

r*x  .0 

9-9640 

0-9205 

11  16 

169 

12 

0  48 

2-7 

9-5902 

0-3893 

5  9-2 

25  3 

f)X.9 

9-9643 

0-9211 

11  12 

168 

13 

0  52 

7-5 

9-5885 

0-3877 

5  34-4 

ZO  Zi 

OX  .A 

9-9646 

0-92  IS 

11  8 

167 

14 

0  56 

12-3 

9-5867 

0-3861 

5  59-4 

ZO  U 

24-9 

9-9649 

0-9224 

11  4 

166 

15 

1  0 

16-9 

9-5848 

0-3844 

6  24-3 

C\A  .O 

9-9653 

0-9232 

11  0 

165 

16 

1  4 

21-4 

9-5827 

0-3825 

6  49-1 

2A  o 

9/1  .fi 

9-9656 

0-9239 

10  56 

164 

17 

1  8 

25-9 

9-5804 

0-3806 

7  13-7 

124  O 

CIA  *A 

9-9660 

0-9247 

10  52 

163 

18 

1  12 

30-2 

9-5780 

0-3785 

7  38-1 

J4  4 
04.  -Q 

9-9664 

0-9256 

10  48 

162 

19 

1  16 

34-3 

9-5755 

0-3763 

8  2-3 

.£4  £ 

24-0 

9-9668 

0-9265 

10  44 

161 

20 

1  20 

38-4 

9-5728 

0-3740 

8  26-3 

OO.Q 

9-9673 

0-9274 

10  40 

160 

21 

1  24 

42-3 

9-5700 

0-3715 

8  50-2 

2.6  y 

9O  .ft 

9-9677 

0-9284 

10  36 

159 

22 

1  28 

46  1 

9-5670 

0-3690 

9  13-8 

Zi6  O 

OO  •  1 

9-9682 

0-9294 

10  32 

158 

23 

1  32 

49-8 

9-5639 

0-3663 

9  37-2 

2.6  4 
90  .9 

9  -9687 

0-9305 

10  28 

157 

24 

1  36 

53-3 

9-5606 

0-3635 

10  0-4 

Z6  2. 

22-9 

9-9692 

0-9316 

10  24 

156 

25 

1  40 

56-6 

9-5571 

0-3607 

10  23-3 

O9-R 

9-9697 

0-9327 

10  20 

155 

26 

1  44 

59-8 

9-5535 

0-3577 

10  45-9 

2,6  O 
09.  x 

9-9703 

0-9338 

10  16 

154 

27 

1  48 

2  2-9 

9-5497 

0-3546 

11  8-4 

££  O 
99.9 

9-9708 

0-9350 

10  12 

153 

28 

1  52 

2  5-7 

9-5458 

0-3514 

11  30-6 

__  z 

09  .A 

9-9714 

0-9362 

10   8 

152 

29 

1  56 

2  8-5 

9-5416 

0-3481 

11  52-6 

2.Z,  U 

21-6 

9-9720 

0-9375 

10  4 

151 

30 

2  0 

2  11-0 

9-5374 

0-3446 

12  14-2 

91  -A. 

9-9725 

0-9387 

10  0 

150 

31 

2  4 

2  13-4 

9-5329 

0-3411 

12  35-6 

2ii  4 

91  •  1 

9-9731 

0-9400 

9  56 

149 

32 

2  8 

2  15-6 

9-5282 

0-3375 

12  56-7 

-  1  1 

OA  .Q 

9-9737 

0-9413 

9  52 

148 

33 

2  12 

2  17-6 

9-5234 

0-3337 

13  17-5 

2\J  O 
9A.fi 

9-9744 

0-9427 

9  48 

147 

34 

2  16 

2  19-5 

9-5184 

0-3299 

13  38-1 

ZU  D 

20-2 

99750 

0-9440 

9  44 

146 

35 

2  20 

2  21-2 

9-5132 

0-3260 

13  58-3 

1  Q-Q 

9-9756 

0-9454 

9  40 

145 

36 

2  24 

2  22-7 

9-5078 

0-3220 

14  18-2 

iy  y 

1  f\  •  (* 

9-9762 

0-9468 

9  36 

144 

37 

2  28 

2  24-0 

9-5022 

0-3178 

14  37-8 

iy  o 

I  Q.O 

9-9769 

0-9482 

9  32 

143 

38 

2  32 

2  25-1 

9-4964 

0-3136 

14  57-1 

iy  o 

i  n  .A 

9-9775 

0-9496 

9  28 

142 

39 

2  36 

2  26-1 

9-4903 

0-3093 

15  16-1 

iy  u 

18-7 

9-9782 

0-9510 

9  24 

141 

40 

2  40 

2  26-8 

9-4841 

0-3048 

15  34-8 

1  o  .q 

9-9788 

0-9524 

9  20 

140 

41 

2  44 

2  27-4 

9-4776 

0-3003 

15  53-1 

lo  o 
1  S  -H 

9-9795 

0-9538 

9  16 

139 

42 

2  48 

2  27-8 

9-4709 

0-2957 

16  ll'l 

lo  U 

n.rr 

9-9801 

0-9553 

9  12 

138 

43 

2  52 

2  28-1 

9-4640 

0-2910 

16  28-8 

7 
17.0 

9-9808 

0-9567 

9  8 

137 

44 

2  56 

2  28-1 

9-4568 

0-2863 

16  46-1 

1  /  u 

17-0 

.9-9814 

0-9581 

9  4 

136 

45 

3  0 

2  28-0 

9-4493 

0-2814 

17  3-1 

9-9821 

0-9596 

9  0 

135 

APPENDIX   VI. 

TABLE  XX.— Concluded. 

Conversion  of  Longitude  and  Latitude  into 

Declination,  and  vice  versa. 

e  =  23°  21'  0". 


431 


Ascension  and 


k. 

L 

A. 

log  a. 

a. 

B. 

Diff. 

log  b. 

b. 

k. 

k. 

0 

h.  rn. 

0       / 

, 

h.  m. 

0 

45 

3  0 

2  28-0 

9-4493 

0-2814 

17  3-1 

lfi-7 

9-9821 

0-9596 

9 

135 

46 

3  4 

2  27-7 

9-4416 

0-2764 

17  19-8 

ID  / 

1  f>  .0 

9-9827 

0-9610 

8  56 

134 

47 

3  8 

2  27  "2 

9-4336 

0-2714 

17  36-1 

JrJ.q   9-9834 

0-9625 

8  52 

133 

48 

3  12 

2  26-5 

9-4253 

0-2663 

17  52-0 

I  K  .(X 

9-9840 

0-9639 

8  48 

132 

49 

3  16 

2  25-7 

9-4168 

0-2611 

18  7-6 

ID  O 

15-3 

9-9847 

0-9653 

8  44 

131 

50 

3  20 

2  24-7 

9-4079 

0-2558 

18  22-9 

I  A  ,{\ 

9-9853 

0-9667 

8  40 

130 

51 

3  24 

2  23-5 

9-3987 

0-2504 

18  37-8 

14:  y 

I  A.K 

9-9859 

0-9681 

8  36 

129 

52 

3  28 

2  22-1 

9-3892 

0-2450 

18  52-3 

14:  O 

14.  '9 

9-9866 

0-9695 

8  32 

128 

53 

8  32 

2  20-6 

9-3793 

0-2395 

19  6-5 

IT:  ^ 

i  q.o 

9-9872 

0-9709 

8  28 

127 

54 

3  36 

2  18-9 

9-3690 

0-2339 

19  20-3 

lo  o 
13-4 

9-9878 

0-9723 

8  24 

126 

•  55 

3  40 

2  17-1 

9-3584 

0-2283 

19  33-7 

1  O  .] 

9-9884 

0-9736 

8  20 

125 

56 

3  44 

2  15-1 

9-3474 

0-2225 

19  46-8 

I  o  1 

9-9890 

0-9749 

8  16 

124 

57 

3  48 

2  12-9 

9-3359 

0-2167 

19  59-5 

o!q 

9-9896 

0-9762 

8  12 

123 

58 

3  52 

2  10-6 

9-3240 

0-2109 

20  11-8 

1  9-n 

9-9901 

0-9775 

8  8 

122 

59 

3  56 

2  8-1 

9-3117 

0-2050 

20  23-8 

\£i  U 

11-6 

9-9907 

0-9788 

8  4 

121 

60 

4  0 

2  5-5 

9-2988 

0-1990 

20  35-4 

U«ti 

9-9912 

0-9800 

8  0 

120 

61 

4  4 

2  2-7 

9-2854 

0-1929 

20  46-6 

& 

9-9918 

0-9812 

7  56 

119 

62 

4  8 

1  59-8 

9-2714 

0-1868 

20  57-4 

10*8 

1  fl'^i 

9-9923 

0-9824 

7  52 

118 

63 

4  12 

1  56-8 

9-2569 

0-1807 

21  7-9 

1U  O 

9  -9928 

0-9885 

7  48 

117 

64 

4  16 

1  53-6 

9-2417 

0-1744 

21  18-0 

IQ'.]   9-9933 

0-9847 

7  44 

116 

65 

4  20 

1  50-3 

9-2258 

0-1682 

21  27-7 

9.0 

9-9938 

0-9858 

7  40 

115 

66 

4  24 

1  46-9 

9-2091 

0-1619 

21  37-0 

A 

Q-n 

9-9942 

0-9868 

7  36 

114 

67 

4  28 

1  43-4 

9-1917 

0-1555 

21  46-0 

y  u 

C  .£» 

9-9947 

0-9878 

7  32 

113 

68 

4  32 

1  39-7 

9-1734 

0-1491 

21  54-6 

o  D 
8  "9 

99951 

0-9888 

7  28 

112 

69 

4  36 

1  36-0 

9-1542 

0-1426 

22  2-8 

Z 

7-8 

9-9955 

0-9898 

7  24 

111 

70 

4  40 

1  32-1 

9-1339 

0-1361 

22  10-6 

7.4 

9-9959 

0-9907 

7  20 

110 

71 

4  44 

1  28-2 

9-1125 

0-1296 

22  18-0 

4: 

9-9963 

0-9916 

7  16 

109 

72 

4  48 

1  24-1 

9-0898 

0-1230 

22  25-1 

i  "1 

6-7 

9-9967 

0-9924 

7  12 

108 

73 

4  52 

1  19-9 

9-0658 

0-1164 

22  31-8 

/ 

6.0 

9-9970 

0-9932 

7  8 

107 

74 

4  56 

1  15-7 

9-0402 

0-1097 

22  38-1 

«1 

5-9 

9-9974 

0-9940 

7  4 

106 

75 

5  0 

1  11-4 

9-0128 

0-1030 

22  44-0 

5-ig 

9-9977 

0-9947 

7  0 

105 

76 

5  4 

1  7-0 

8-9835 

0-0963 

22  49-5 

o 

K.O 

9-9980 

0-9954 

6  56 

104 

77 

5  8 

1  2-5 

8-9519 

0-0895 

22  54-7 

O  ^ 

4.0 

9-9983 

0-99HO 

6  52 

103 

78 

5  12 

0  57-9 

8-9177 

0-0827 

22  59-5 

O 

A'A. 

9-9985 

0-9966 

6  48 

102 

79 

5  16 

0  53-3 

8-8804 

0-0759 

•J3  3-9 

4:  4: 

4-0 

9-9987 

0-9971 

6  44 

101 

80 

5  20 

0  48-7 

8-8395 

0-0691 

23  7-9 

q  .a 

9-9990 

0-9976 

6  40 

100 

81 

5  24 

0.43-9 

8-7942 

0-0623 

23  11-5 

o  O 

.  >  .*> 

9-9992 

0-9981 

6  36 

99 

82 

5  28 

0  39-2 

8-7434 

0-0554 

23  14-8 

33 

Q.O 

9-9993 

0-9985 

6  32 

98 

83 

5  32 

0  34-4 

8-6857 

0-0485 

23  17-6 

A  O 

9-9995 

0-9988 

6  28 

97 

84 

5  36 

0  29-5 

8-6191 

0-0416 

23  20-1 

2-1 

9-9996 

0-9991 

6  24 

96 

85 

5  40 

0  24-7 

8-5401 

0-0347 

23  22-2 

1B*f 

9-9997 

0-9994 

6  20 

95 

86 

5  44 

0  19-8 

8-4434 

0-0278 

23  23-9 

7 

1  .4 

9-9998 

0-9996 

6  16 

94 

87 

5  48 

0  14-8 

8-3186 

0-0208 

23  25-3 

1  4: 
O.Q 

9-9999 

0-9998 

6  12 

93 

88 

5  52 

0  9-9 

8-1426 

0-0139 

23  26-2 

y 

n.c 

o-oooo 

0-9999 

6  8 

92 

89 

5  56 

0  5-0 

7-8417 

0-0069 

23  26-8 

U  O 

0-2 

o-oooo 

1  -0000 

6  4 

91 

90 

6  0 

0  0-0 

-  00 

o-oooo 

23  27-0 

o-oooo 

1  -0000 

6  0 

90 

432 


APPENDIX  VI. 
TABLE    XXI. 


Factors  for  converting  Small  Changes  of  Latitude  and  Longitude  near  the 
Ecliptic  into  Changes  of  Eight  Ascension  and  Declination. 


Formulae  :  5a  = 


va)  dv  +  (/3a)  5/3, 


(va). 

(/3a) 

' 

Longi- 
*  „  j 

foft 

(88) 

tude. 

*---r. 

0°. 

+  5°. 

,--  r. 

0°. 

+5°. 

\  vu  j* 

0 

270 

+  •133  + 

+  •090  + 

+  •050  + 

•ooo 

•ooo 

•ooo 

•ooo 

•ooo 

o 
270 

275 

•131 

•089 

•049 

-•043  + 

-  -041  + 

-  '040  + 

+  -038  - 

-  -001  - 

265 

280 

•126  + 

•084  + 

•045  + 

-•085  + 

-•081  + 

-•079  + 

•075- 

-  -003  - 

260 

285 

+  •117 

+  •076 

+  -039 

-  -126 

-  -121 

-•117 

+  •111- 

-•006- 

255 

290 

•105 

•066 

•030 

-•165 

-•158 

-  -154 

•147- 

-•011- 

250 

295 

•091  + 

•055  + 

•021  + 

-  -202  + 

-•193  + 

-•188  + 

•180- 

-  -016  -     245 

300 

+  •074 

+  •041 

+  •009  + 

-•235 

-•226 

-•220 

+  -212  - 

-  -023  - 

240 

305 

•057 

•027 

-  -003  - 

-  -265 

-  -256 

-  -250 

•242- 

-•030- 

235 

310 

•039  + 

+  •011  + 

-•016- 

-•292  + 

-  -282  + 

-•277  + 

•269- 

-•037- 

230 

315 

+  •021 

-  -004  - 

-  -028  - 

-•316 

-•306 

-•301 

+  •293- 

-  -044  - 

225 

320 

+  •003  + 

-  -018  - 

-  -040  - 

-  -336 

-  -326 

-  -322 

•315- 

-  -051  - 

220 

325 

-  -014  - 

-  -032  - 

-  -052  - 

-•353  + 

-•344  + 

-  '340  + 

•335  - 

-  -058  - 

215 

330 

-•030- 

-  -045  - 

-  -061  - 

-•368 

-  '359 

-•356 

+  '352  - 

-  -064  - 

210 

335 

-•044- 

-  -056  - 

-  -070  - 

-•379 

-•371 

-  -369 

•366- 

-  -069  - 

205 

340 

-  -056  - 

-  -065  - 

-  -077  - 

-  -388  + 

-  -381  + 

-•379  + 

•378- 

-•074- 

200 

345 

-  -066  - 

-  -073  - 

-  -082  - 

-  395 

-•388 

-•388 

+  -386  - 

-  -078  - 

195 

350 

-•074- 

-  -078  - 

-  -085  - 

-•399 

-  -394 

-  -394 

•393  - 

-•080- 

190 

355 

-  -080  - 

-  -081  - 

-  -085  - 

-•401  + 

-•397  + 

-  -399  + 

•397- 

-  -082- 

185 

0 

-  -084  - 

-•083- 

-  -084  - 

-•401 

-•398 

-•401 

+  -398  - 

-  -082  - 

180 

5 

-  -085  - 

-  -081  - 

-•080- 

-  -399 

-  -397 

-•401 

•397  - 

-  -082  - 

175 

10 

-  -085  - 

-  -078  - 

-  -074  - 

-  -394  + 

-•394  + 

-•399  + 

•393- 

-•080- 

170 

15 

-  -082  - 

-  -073  - 

-  -066  - 

-  -388 

-  -388 

-•395 

+  -386  - 

-  -078  - 

165 

20 

-•077- 

-  -065  - 

-  -056  - 

-•379 

-  -381 

-  '388 

•378- 

-•074- 

160 

25 

-  -070  - 

-  -056  - 

-  -044  - 

-•369  + 

-  -371  + 

-•379  + 

•366- 

-  -069  - 

155 

30 

-  -061  - 

-  -045  - 

-•030- 

-•356 

-•359 

-  -368 

+  -352 

-  -064  - 

150 

35 

-  -052  - 

-•032- 

-  -014  - 

-•340 

-•344 

-•353 

•335  - 

-  -058  - 

145 

40 

-  -040  - 

-•018- 

+  •003  + 

-•322  + 

-•326  + 

-•336  + 

•315- 

-  -051  - 

140 

45 

-  -028- 

-  -004  - 

+  •021 

-•301 

-•306 

•316 

+  -293  - 

-  -044  - 

135 

50 

-•016- 

+  •011  + 

•039 

-•277 

-•282 

-  -292 

•269- 

-  -037  - 

130 

55 

-•003- 

+  '027  + 

•057  + 

-•250  + 

-•256  + 

-  -265  + 

•242- 

-  -030  - 

125 

60 

+  •009  + 

+  •041 

+  •074 

-  -220 

-•226 

-•235 

+  -212  - 

-  -023  - 

120 

65 

•021 

•055 

•091 

-  -188 

-•193 

-•202 

•180- 

-•016- 

115 

70 

•030  + 

•066  + 

•105  + 

-•154  + 

-•158  + 

-•165  + 

•147- 

-•011- 

110 

75 

+  -039 

+  •076 

+  •117 

-•117 

-•121 

-•126 

+  •111- 

-•006- 

105 

80 

•045 

•084 

•126 

-•079 

-•081 

-  -085 

•075  - 

-  -003  - 

100 

85 

•049  + 

•089  + 

•131  + 

-•040  + 

-  -041  + 

-  -043  + 

•038  - 

-  -001  - 

95 

90 

+  •050  + 

+  -090  + 

+  •133  + 

•ooo 

•ooo 

•ooo 

•ooo 

•ooo 

90 

0=  -5°. 

0°. 

+5°. 

£=-5°. 

0°. 

+5°. 

Longi- 
tude. 

APPENDIX  VII. 

TABLE   XXII. 

Table  of  Refractions  for  50°  F.  Temp,  and  30  in.  Pressure. 


££: 

Mean 
Refrac- 
tion. 

Change  for 

ipDp: 

Mean 
Refrac- 
tion. 

Change  for 

10°  F. 
Temp. 

1  in.  Bar. 

10°  F. 
Temp. 

1  in.  Bar. 

0 

n 

„ 

// 

o 

/        // 

n 

n 

0 

o-oo 

o-oo 

o-oo 

45 

0  58-2 

-    I'l 

+  2-0 

1 

1-02 

-0-02 

+  0-03 

46 

1     0-2 

-    1-2 

+  2-1 

2 

2-03 

-0-04 

+  0-07 

47 

1     2-4 

-   1-3 

+  2-1 

3 

3-05 

-0-06 

+  0-10 

48 

1     4-6 

-   1-3 

+  2-2 

4 

4-07 

-0-08 

+  0-14 

49 

1     6-9 

-   1-3 

+  2-2 

5 

5-09 

-0-10 

+  0-17 

60 

1     9-3 

-    1-4 

+  2-3 

6 

6-12 

-0-12 

+  0-21 

51 

1    11-8 

-   1-4 

+  2-4 

7 

7-15 

-0-14 

+  0-24 

52 

1    14-4 

-   1-5  • 

+  2-5 

8 

8-18 

-0-16 

+  0-28 

53 

1    17-1 

-   1-5 

+  2-6 

9 

9-22 

-0-18 

+  0-31 

54 

1   20-0 

-    1-6 

-h  2-7 

10 

10-27 

-0-21 

+  0-34 

55 

23-0 

-    1-6 

+  2-8 

11 

11-32 

-0-23 

+  0-38 

56 

26-1 

-    1-7 

+  2-9 

12 

12-38 

-0-25 

+  0-42 

57 

29-4 

-   1-7 

+  3-0 

13 

13-44 

-0-26 

+  0-45 

58 

32-9 

-   1-8 

+  3-2 

14 

14-52 

-0-29 

+  0-49 

59 

36-6 

-   1-9 

+  3-3 

15 

15-60 

-  0-31 

+  0-53 

60 

40-5 

-   2-0 

+  3-4 

16 

16-70 

-0-33 

+  0-56 

61 

44-6 

-   2-1 

+  3-6 

17 

17-80 

-0-35 

+  0-60 

62 

49-1 

-   2-2 

+  3-7 

18 

18-92 

-0-37 

+  0-64 

63 

53-8 

-   2-2 

+  3-9 

19 

20-04 

-0-39 

+  0-68 

64 

58-8 

-   2-3 

+  4-0 

20 

21-19 

-0-42 

+  0-72 

65 

2     4-2 

-   2-4 

+  4-2 

21 

22-35 

-0-44 

+  0-76 

66 

2   10-0 

-   2-5 

+  4-4 

22 

23-52 

-0-46 

+  0-80 

67 

2   16-3 

-   2-6 

+  4-6 

23 

24-71 

-0-48 

+  0-84 

68 

2  23-1 

-   2-8 

+  4-9 

24 

25-92 

-0-51 

+  0-88 

69 

2  30-5 

-   2-9 

+  5-1 

25 

27-15 

-0-54 

+  0-92 

70 

2  38-6 

-   3-1 

+  5-4 

26 

28-39 

-0-56 

+  0-97 

71 

2  47-5 

-   3-3 

+  5-7 

27 

29-66 

-0-58 

+  1-01 

72 

2  57-3 

-   3-5 

+  6-0 

28 

30-95 

-0-60 

+  1-05 

73 

3     8-2 

-  3-7 

+  6-4 

29 

32-26 

-0-63 

+  1-10 

74 

3  20-3 

-   4-0 

+  6-8 

30 

33-60 

-0-65 

+  1-15 

75 

3  33-9 

-   4-3 

+  7-3 

31 

34-97 

-0-69 

+  1-19 

76 

3  49-4 

-   4-6 

+  7-8 

32 

36-37 

-0-72 

+  1-23 

77 

4     7-0 

-   5-0 

+  8-3 

33 

37-79 

-0-74 

+  1-28 

78 

4  27-4 

-   5-5 

+  9-0 

34 

39-26 

-0-77 

+  1-33 

79 

4  51-1 

-   6-0 

+  9-9 

35 

40-75 

-0-80 

+  1-38 

80 

5   19-0 

-   6-5 

+  10-9 

36 

42-28 

-0-83 

+  1-43 

81 

5  52-5 

-   7-3 

+  12-0 

37 

43-84 

-0-86 

+  1-48 

82 

6  33-2 

-   8-2 

+  13-4 

38 

45-46 

-0-89 

+  1-54 

83 

7  23-7 

-  9-3 

+  15-3 

39 

47-12 

-0-92 

+  1-60 

84 

8  27-7 

-10-8 

+  17-5 

40 

48-82 

-0-96 

+  1-66 

85 

9  51-4 

-12-9 

+  20-4 

41 

50-57 

-0-99 

+  1-72 

86 

11   44-3 

-16-0 

+  24-3 

42 

52-37 

-1-02 

+  1-78 

87 

14  22-6 

-20-6 

+  30-0 

43 

54-24 

-1-07 

+  1-84 

88 

18   16-1 

-28-3 

+  38-6 

44 

56-17 

-1-10 

+  1-91 

89 

24  20-6 

-41-7 

+  52-4 

45 

58-16 

-1-14 

+  1-97 

90 

34  32-1 

-68-6 

+  76-5 

2E 


APPENDIX  VIII. 


COEFFICIENTS  FOE  THE  NUTATION  AND  THE  KELATED 
STAR  CONSTANTS. 

0,  Longitude  of  Moon's  node. 

w,  Distance  from  node  to  lunar  perigee. 

g,  Moon's  mean  anomaly. 

d,  Moon's  mean  longitude  =  g  +  w  +  ft. 
D,  Mean  elongation  of  Moon  from  Sun. 
a/,  Distance  from  Moon's  node  to  solar  perigee. 
g',  Sun's  mean  anomaly. 
L,  Sun's  mean  longitude. 
T,  Time  after  1900  in  centuries. 

Each  number  in  the  column  S\j/  is  the  coefficient  of  the  sine  of  the  corresponding 
argument  in  the  expression  of  5^,  and  each  in  column  6e  is  the  coefficient  of  the 
cosine  of  the  argument  in  the  expression  for  the  obliquity.  The  two  remaining 
columns  give  the  corresponding  coefficients  for  the  star  constants,  A  and  B. 

Action  of  the  Moon. 


Argument. 

5f, 

5e. 

A. 

B. 

ft 

sin 

-17-234 

cos 
+  9-210 

sin 
-0-34215 

cos 

-9-210 

-  0-017  T 

+  0-000977 

-0  -00030  T 

-0-000977 

2ft 

+     -209 

-    -090 

+  -00415 

+    '090 

2« 

-      -204 

+    -089 

-  -00405 

-    -089 

g 

+     '068 

•000 

+  -00135 

•000 

2d  -ft 

-      -034 

+    -018 

-  -00068 

-    -018 

2«+0 

-      -026 

+  -on 

-  -00052 

-    -Oil 

2D-fl 

+     -015 

•000 

+  -00030 

•ooo 

2L-Q 

+     -012 

-    '007 

+  -00025 

+   -007 

2(1-0 

+    -on 

-    -005 

+  -00023 

+    '005 

2D 

+     -006 

•000 

+  -00012 

•ooo 

0  +  ft 

+     -006 

-    -003 

+  -00012 

+    -003 

-g+Q 

•006 

+    -003 

-  -00011 

-    -003 

2w  +  ft 

+     -005 

-    -003 

+  -00010 

+    -003 

2D  +  2d-g 

-      '005 

+    '002 

-  -oooio 

-    -002 

2D-2g 

-      -004 

•000 

-•00009 

•ooo 

2«+0-ft 

-      -004 

+    -002 

-  -00009 

-    -002 

2Z>  +  2« 

-      -003 

•000 

-  -00006 

•000 

20 

+     -003 

•000 

+  -00006 

•ooo 

2(1+20 

-      -003 

•000 

-  -00005 

•ooo 

2L+g 

+     -003 

•000 

+  -00005 

•ooo 

2L 

-      -002 

•ooo 

-  -00004 

•ooo 

Action  of  the  Sun. 


Argument. 

ty. 

5e. 

A. 

B. 

2L 

2L+g' 
ZL-g' 

sin 

-1-270 
+    -128 
-    -050 
+    -021 

cos 
+  0-551 

•ooo 

+    -022 
-    -009 

sin 

n 

-0-02521 
+    -00254 
-    -00099 
+  0-0042 

cos 

-0-551 
•000 
-    -022 
+    -009 

APPENDIX  IX. 

TABLE   XXIII. 

Three-Place  Logarithms^ 


1 

2 

•000 
•301 

301 

51 
52 

•708 
•716 

8 

101 
102 

•004 
•009 

5 

151 

152 

•179 
•182 

3 

3 

•477 

125 

53 

•724 

8 

103 

•013 

4 

153 

•185 

3 

4 

•602 

07 

54 

•732 

8" 

104 

•017 

A 

154 

•188 

5 

•699 

«'  / 

1Q 

55 

•740 

105 

•021 

rr 

155 

•190 

6 

•778 

/y 

Q*1 

56 

•748 

106 

•025 

156 

•193 

b/ 

8 

4 

7 

•845 

CO 

57 

•756 

107 

•029 

157 

•196 

8 

•903 

Oo 
ei 

58 

•763 

7 

108 

•033 

4 

158 

•199 

9 

•954 

Ol 

59 

•771 

109 

•037 

: 

159 

•201 

46 

7 

4 

3 

W 

•000 

41 

60 

•778 

7 

110 

•041 

4 

160 

•204 

3 

11 

•041 

OO 

61 

•785 

111 

•045 

161 

•207 

12 

•079 

OO 

62 

•792 

7 

112 

•049 

4 

162 

•210 

13 

•114 

35 

OO 

63 

•799 

7 

113 

•053 

4 

163 

•212 

2 

.32 

7 

4 

14 

•146 

on 

64 

•806 

I_ 

114 

•057 

164 

•215 

15 

•176 

ou 

65 

•813 

j 

115 

•061 

165 

•217 

16 

•204 

O/* 

66 

•820 

7 

116 

•064 

3 

166 

•220 

26 

6 

4 

3 

17 

•230 

OK 

67 

•826 

117 

•068 

167 

•223 

18 

•255 

iso 

68 

•833 

7 

118 

•072 

4 

168 

•225 

19 

•279 

22 

69 

•839 

6 

119 

•076 

3 

169 

•228 

2 

20 

•301 

r\  i 

70 

•845 

120 

•079 

170 

•230 

21 

6 

4 

3 

21 

•322 

71 

•851 

121 

•083 

171 

•233 

22 
23 

•342 
•362 

20 
20 
18 

72 
73 

•857 
•863 

6 
6 
6 

122 
123 

•086 
•090 

3 
4 
3 

172 
173 

•236 
•238 

3 
2 
3 

24 

•380 

1  C 

74 

•869 

124 

•093 

174 

•241 

25 

•398 

io 

75 

•875 

6 

125 

•097 

175 

•243 

26 

•415 

76 

•881 

126 

•100 

176 

•246 

16 

5 

4 

27 

•431 

77 

•886 

127 

•104 

177 

•248 

28 

•447 

16 

IK 

78 

•892 

6 

128 

•107 

3 

178 

•250 

29 

•462 

JIO 

79 

•898 

129 

•111 

179 

•253 

_ 

15 

5 

3 

30 

•477 

14 

80 

•903 

5 

130 

•114 

3 

180 

•255 

3 

31 

•491 

81 

•908 

131 

•117 

181 

•258 

32 

•505 

14 

82 

•914 

6 

132 

•121 

4 

182 

•260 

33 

•519 

14 

83 

•919 

5 
5 

133 

•124 

3 
3 

183 

•262 

3 

34 

•531 

1  O 

84 

•924 

134 

•127 

184 

•265 

35 

•544 

13 

19 

85 

•929 

5 

135 

•130 

3 

185 

•267 

36 

•556 

125 

12 

86 

•934 

5 
6 

136 

•134 

3 

186 

•270 

2 

37 

•568 

87 

•940 

137 

•137 

187 

•272 

38 

•580 

12 

88 

•944 

4 

138 

•140 

q 

188 

•274. 

39 

•591 

89 

•949 

139 

•143 

o 

189 

•276 

11 

5 

3 

40 

•602 

11 

90 

•954 

5 

140 

•146 

3 

190 

•279 

2 

41 

•613 

i  f\ 

91 

•959 

141 

•149 

191 

•281 

42 

•623 

10 

92 

•964 

5 

142 

•152 

192 

•283 

43 

•633 

10 
10 

93 

•968 

4 
5 

143 

•155 

3 

193 

•286 

2 

44 

•643 

i  n 

94 

•973 

144 

•158 

194 

•288 

45 

•653 

1U 

1  A 

95 

•978 

5 

145 

•161 

195 

•290 

46 

•663 

1U 

9 

96 

•982 

4 
5 

146 

•164 

3 

196 

•292 

2 

47 

•672 

97 

•987 

147 

•167 

q 

197 

•294 

48 

•681 

98 

•991 

4 

148 

•170 

0 

198 

•297 

49 

•690 

9 
9 

99 

•996 

5 

4 

149 

•173 

3 

199 

•299 

2 

50 

•699 

100 

•000 

150 

•176 

200 

•301 

436 


APPENDIX   IX. 

TABLE   XXIV. 

Logarithmic  Sines,  etc.,  for  every 


• 

Sin. 

Diff. 

Tan. 

Diff. 

Cot. 

Cos. 

• 

0 

90 

1 

2 
3 

8-242 
8-543 
8-719 

301 
176 
125 

8-242 
8-543 
8-719 

301 
176 
126 

1-758 
1-457 
1-281 

o-ooo 
o-ooo 

9-999 

89 
88 
87 

4 

8-844 

8-845 

1-155 

9-999 

86 

5 

8-940 

•7A 

8-942 

OA 

1-058 

9-998 

85 

6 

9-019 

/y 

67 

9-022 

oO 

67 

0-978 

9-998 

84 

7 

9-086 

9-089 

0-911 

9-997 

83 

8 

9-144 

Oo 

9-148 

OJ 

0-852 

9-996 

82 

9 

9-194 

50 

AC 

9-200 

52 

A  O 

0-800 

9-995 

81 

46 

46 

10 

9-240 

41 

9-246 

43 

0-754 

9-993 

80 

11 

9-281 

f\rj 

9-289 

OO 

0-711 

9-992 

79 

12 

9-318 

ol 
04 

9-327 

.38 

Of" 

0-673 

9-990 

78 

13 

9-352 

O1* 

32 

9-363 

oo 
34 

0-637 

9-989 

77 

14 

9-384 

9-397 

01 

0-603 

9-987 

76 

15 

9-413 

97 

9-428 

ol 

0-572 

9-985 

75 

16 

9-440 

mi 

26 

9-457 

28 

0-543 

9-983 

74 

17 

9-466 

9-485 

97 

0-515 

9-981 

73 

18 

9-490 

90 

9-512 

*/ 

nx 

0-488 

9-978 

72 

19 

9-513 

BO 

21 

9-537 

400 

24 

0-463 

9-976 

71 

20 

9-534 

20 

9-561 

23 

0-439 

9-973 

70 

21 

9-554 

9-584 

99 

0-416 

9-970 

69 

22 

9-574 

^0 

9-606 

ifiS 

0-394 

9-967 

68 

23 

9-592 

18 
17 

9-628 

22 

21 

0-372 

9-964 

67 

24 

9-609 

9-649 

0-351 

9-961 

66 

25 

9-626 

i  <? 

9-669 

20 

0-331 

9-957 

65 

26 

9-642 

lo 
15 

9-688 

19 

0-312 

9-954 

64 

27 

9-657 

9-707 

0-293 

9-950 

63 

28 

9-672 

15 

9-726 

19 

I  0 

0-274 

9-946 

62 

29 

9-686 

13 

9-744 

lo 

17 

0-256 

9-942 

61 

30 

9-699 

13 

7-761 

18 

0-239 

9-938 

60 

31 

9-712 

9-779 

0-221 

9-933 

59 

32 

9-724 

19 

9-796 

0-204 

9-928 

58 

33 

9-736 

V6 

i  ,  * 

9-813 

. 

0-187 

9-924 

57 

12 

16 

34 

9-748 

9-829 

0-171 

9-919 

56 

35 

9-759 

11 

1  A 

9-845 

16 

1  C* 

0-155 

9-913 

55 

36 

9-769 

10 
10 

9-861 

16 
16 

0-139 

9-908 

54 

37 

9-779 

1  A 

9-877 

1  £? 

0-123 

9-902 

53 

38 

9-789 

10 

1  II 

9-893 

16 

I  E 

0-107 

9-897 

52 

39 

9-799 

1U 

9 

9-908 

1O 

16 

0-092 

9-891 

51 

40 

9-808 

9 

9-924 

15 

0-076 

9-884 

50 

41 

9-817 

9-939 

1  e 

0-061 

9-878 

49 

42 

9-826 

9-954 

JLO 

i  a 

0-046 

9-871 

48 

43 

9-834 

8 

9-970 

16 
15 

0-030 

9-864 

47 

44 

9842 

9-985 

1  C 

0-015 

9-857 

46 

45 

9-849 

o-ooo 

15 

o-ooo 

9-849 

45 

0 

Cos. 

Cot. 

Tan. 

Sin. 

0 

APPENDIX   IX. 


437 


TABLE  XXV. 

Logarithmic  Sines,  etc.,  for  every  Tenth  of  a  Degree  to  5°. 


Sin. 

Diff. 

Tan. 

Diff. 

Cot. 

Cos. 

• 

00 

o-ooo 

90-0 

O'l 
0*2 
0-3 
0-4 

7-242 
7-543 
7-719 

7-844 

301 
176 
125 

i\~ 

7-242 
7-543 
7-719 

7-844 

301 
176 
125 

2-758 
2-457 
2-281 
2-156 

o-ooo 
o-ooo 
o-ooo 
o-ooo 

89-9 
89*8 
89-7 
89-6 

0-5 

7-941 

97 
79 

7*941 

97 
79 

2-059 

o-ooo 

895 

0-6 

8-020 

,  .  — 

8-020 

1-980 

o-ooo 

89-4 

0'7 

8-087 

67 

KG 

8-087 

67 

KO 

1-913 

o-ooo 

89-3 

0-8 

8-145 

Oo 

8-145 

DO 

1-855 

o-ooo 

89*2 

0'9 

8-196 

51 

A  £* 

8-196 

51 

1-804 

o-ooo 

89-1 

1-0 

8-242 

46 
41 

8-242 

46 
41 

1-758 

o-ooo 

89-0 

1-1 

8-283 

00 

8-283 

oo 

1-717 

o-ooo 

88-9 

1-2 

8-321 

Oo 

8-321 

oo 

1-679 

o-ooo 

88-8 

1-3 

8-356 

35 

OO 

8-356 

35 

1-644 

o-ooo 

88-7 

1-4 

8-388 

32 

OA 

8-388 

32 

€\f\ 

1-612 

o-ooo 

88-6 

1-5 

8-418 

oU 

28 

8-418 

30 

28 

1-582 

o-ooo 

88-5 

1-6 

8-446 

f)C 

8-446 

9ft 

1-554 

o-ooo 

88*4 

1-7 

8-472 

290 

OK 

8-472 

ZO 

QK 

1-528 

o-ooo 

88-3 

1-8 

8-497 

TSSt 

94. 

8-497 

iiD 

f)A 

•503 

o-ooo 

88-2 

1-9 

8-521 

^^r 

8-521 

At 

•479 

o-ooo 

88-1 

2-0 

8-543 

22 
21 

8-543 

22 
21 

•457 

o-ooo 

88-0 

21 

8-564 

9n 

8-564 

91 

•436 

o-ooo 

87-9 

2'2 

8-584 

£\) 

i  f\ 

8-585 

25i 

•415 

o-ooo 

87-8 

2-3 

8-603 

19 
i  f\ 

8-604 

19 

1  O 

•396 

o-ooo 

87-7 

2*4 

8-622 

iy 

8-622 

18 

•378 

o-ooo 

87-6 

2'5 

8-640 

18 
17 

8-640 

18 
17 

•360 

o-ooo 

87-5 

2-6 

8-657 

ifi 

8-657 

n- 

•343 

o-ooo 

87-4 

2-7 

8-673 

10 

1  £J 

8-674 

. 

i  e 

•326 

o-ooo 

87-3 

2'8 

8-689 

lo 

8-689 

15 

•311 

9-999 

87-2 

2-9 

8-704 

15 

IK 

8-705 

16 
id 

1-295 

9-999 

871 

3*0 

8-719 

10 
14 

8-719 

ATC 

15 

1-281 

9-999 

87-0 

3-1 

8-733 

8-734 

10 

1-266 

9-999 

86-9 

3'2 

8-747 

10 

8-747 

J.O 

1  A. 

1-253 

9-999 

86-8 

3-3 

8-760 

JLo 

i  •> 

8-761 

14 

1  O 

1-239 

9-999 

86-7 

3-4 

8-773 

lo 

10 

8-774 

13 

19 

1-226 

9-999 

86-6 

3'6 

8786 

lo 

12 

8-786 

IZ 

13 

1-214 

9-999 

865 

3-6 

8-798 

19 

8-799 

19 

1-201 

9-999 

86-4 

3-7 

8-810 

\.& 

8-811 

JL2C 

•189 

9-999 

86-3 

3'8 

8-821 

11 
19 

8-822 

11 

19 

•178 

9-999 

86-2 

3'9 

8-833 

\M 

8-834 

i.^ 

•166 

9-999 

86-1 

4-0 

8-844 

. 

10 

8-845 

10 

•155 

9-999 

86-0 

4-1 

8-854 

8-855 

•145 

9-999 

85-9 

4'2 

8-865 

8-866 

•134 

9-999 

85-8 

4*3 

8-875 

10 
in 

8-876 

10 
in 

•124 

9-999 

85-7 

4-4 

8-885 

1U 

8-886 

1U 

•114 

9-999 

85-6 

4*5 

8-895 

10 
9 

8-896 

10 
10 

•104 

9-999 

85-5 

4-6 

8-904 

8-906 

1-094 

9-999 

85-4 

4'7 

8-913 

in 

8-915 

1-085 

9-999 

85-3 

4-8 

8-923 

1U 

8-924 

1-076 

9-998 

85-2 

4-9 

8-932 

8-933 

1-067 

9-998 

85-1 

5-0 

8-940 

8-942 

. 

1-058 

9-998 

85-0 

° 

Cos. 

Diff. 

Cot. 

Diff. 

Tan. 

Sin. 

0 

APPENDIX   IX. 

TABLE    XXVI. 

Natural  Values  of  the  Trigonometrical  Functions. 


Sin. 

Diflf. 

Tan. 

Diff. 

Sec. 

Diflf. 

o 
0 
1 
2 
3 

•ooo 

•017 
•035 
•052 

17 

18 
17 

•ooo 

•017 
•035 
•052 

17 
18 
17 

1-000 
1-000 
1-001 
1-001 

0 
1 
0 

0 

90 
89 
88 
87 

4 
6 

•070 
•087 

18 
17 

18 

•070 
•087 

18 
17 
18 

1-002 
1-004 

1 
2 
2 

86 
85 

6 
7 
8 
9 
10 

•105 
•122 
•139 
•15t> 
•174 

17 
17 
17 
18 
17 

•105 
•123 
•141 
•158 
•176 

18 
18 
17 

18 
18 

1-006 
1-008 
1-010 
1-012 
1-015 

2 
2 
2 
3 
4 

84 
83 
82 
81 
80 

11 
12 
13 
14 

•191 

•208 
•225 
•242 

17 
17 
17 

•194 
•213 
•231 
•249 

19 
18 
18 

1-019 
1-022 
1-026 
1-031 

3 

4 
5 

79 
78 
77 
76 

15 

•259 

17 
17 

•268 

19 
19 

1-035 

4 
5 

75 

16 
17 
18 

•276 
•292 
•309 

16 
17 

•287 
•306 
•325 

19 
19 

1-040 
1-046 
1-051 

6 
5 

74 
73 
'72 

19 
20 

•326 
•342 

17 
16 
16 

•344 
•364 

19 
20 

20 

1-058 
1-064 

7 
6 

7 

71 
70 

21 

•358 

•384 

1-071 

69 

22 
23 
24 
25 

•375 
•391 
•407 
•423 

16 
16 
16 
15 

•404 
•424 
•445 
•466 

20 
21 
21 
22 

1-079 
1-086 
1-095 
1-103 

7 
9 
8 
10 

68 
67 
66 
65 

26 
27 
28 
29 
30 

•438 
•454 
•469 

•485 
•500 

16 
15 
16 
15 
15 

•488 
•510 
•532 
•554 
•577 

22 
22 
22 
23 
24 

1-113 
1-122 
1-133 
1-143 
1-155 

9 
11 
10 
12 
12 

64 
63 
62 
61 
60 

31 
32 
33 
34 
35 

515 
•530 
•545 
•559 
•574 

15 
15 
14 
15 
14 

•601 
•625 
•649 
•675 
•700 

24 
24 
26 
25 
27 

1-167 
1-179 
1-192 
1-206 
1-221 

12 
13 
14 
15 
15 

59 
58 
57 
56 
55 

36 
37 
38 
39 
40 

•588 
•602 
•616 
•629 
•643 

14 
14 
13 
14 
13 

•727 
•754 
•781 
•810 
•839 

27 
27 
29 
29 
30 

1-236 
1-252 
1-269 

1-287 
1-305 

16 
17 
18 
18 
20 

54 
63 
52 
51 
50 

41 
42 
43 
44 
45 

•656 
•669 
•682 
•695 
•707 

13 
13 
13 
12 

•869 
•900 
•933 
•966 
1-000 

31 
33 
33 
34 

•325 
•346 
•367 
•390 
•414 

21 
21 
23 
24 

49 
48 
47 
46 
45 

Cos. 

Diflf. 

Cot. 

Diff. 

Cosec. 

Diff. 

APPENDIX   IX. 

TABLE   XXVL— Concluded. 
Natural  Values  of  the  Trigonometrical  Functions. 


439 


Cosec. 

Diff. 

Cot. 

Diff. 

Cos. 

Diff. 

0 

0 

1-000 

o 
90 

1 
2 
3 
4 

57-299 
28-654 
19-107 
14-336 

28-6 
9-94 

4-77 

O'Sfi 

57-290 
28-636 
19-081 
14-301 

28-6 
9-55 

4-78 
2.07 

1-000 
•999 
•999 
•998 

0 

1 

0 

1 

89 
88 
87 
86 

5 

11-474 

1-91 

11-430 

o/ 

1-92 

•996 

1 

85 

6 
7 
8 
9 
10 

9-567 
8-206 
7-185 
6-392 
5-759 

1-36 
1-02 
•793 
•633 
•518 

9-514 
8-144 
7-115 
6-314 
5-671 

1-37 
1-03 
•801 
•643 
•526 

•995 
•993 
•990 
•988 
•985 

2 

3 
2 

3 
3 

84 
83 
82 
81 
80 

11 
12 
13 
14 
15 

5-241 
4-810 
4-445 
4-134 
3-864 

•431 
•365 
•311 
•270 
•236 

5-145 
4-705 
4-331 
4-011 
3-732 

•440 
•374 
•320 
•279 
•245 

•982 
•978 
•974 
•970 
•966 

4 
4 
4 
4 
5 

79 
78 
77 
76 
75 

16 
17 
18 
19 
20 

3-628 
3-420 
3-236 
3-072 
2-924 

•208 
•184 
•164 
•148 
•134 

3-487 
3-271 
3-078 
2-904 
2-747 

•216 
•193 
•174 
•157 
•142 

•961 
•956 
•951 
•946 
•940 

5 
5 
5 
6 
6 

74 
73 
72 
71 
70 

21 
22 
23 
24 
25 

2-790 
2-669 
2-559 
2-459 
2-366 

•121 
•110 
•100 
•093 

•085 

2-605 
2-475 
2-356 
2-246 
2-145 

•130 
•119 
•110 
•101 
•095 

•934 
•927 
•921 
•914 
•906 

7 
6 

7 
8 
7 

69 
68 
67 
66 
65 

26 
27 
28 
29 
30 

2-281 
2-203 
2-130 
2-063 
2-000 

•078 
•073 
•067 
•063 
•058 

2-050 
1  -963 
1-881 
1-804 
1-732 

•087 
•082 
•077 
•072 
•068 

•899 
•891 

•883 
•875 
•866 

8 
8 
8 
9 
9 

64 
63 
62 
61 
60 

31 
32 
33 
34 
35 

1-942 

•887 
•836 
•788 
•743 

•055 
•051 
•048 
•045 
•042 

1-664 
1-600 
1-540 
1-483 
1-428 

•064 
•060 
•057 
•055 
•052 

•857 
•848 
•839 
•829 
•819 

9 
9 
10 
10 
10 

59 
58 
57 
66 
55 

36 
37 
38 
39 
40 

•701 
•662 
•624 
•589 
•556 

•039 
•038 
•035 
•033 
•032 

1-376 
1-327 
1-280 
1-235 
1-192 

•049 
•047 
•045 
•043 
•042 

•809 
•799 

•788 
•777 
•766 

10 
11 
11 
11 
11 

54 
53 
52 
51 
50 

41 
42 
43 
44 
45 

•524 
•494 
•466 
•440 
1-414 

•030 
•028 
•026 
•026 

1-150 
1-111 
1-072 
1-036 
1-000 

•039 
•039 
•036 
•036 

•755 
•743 
•731 
•719 

•707 

12 
12 
12 
12 

49 
48 
47 
46 
45 

Sec. 

Diff. 

Tan. 

Diff. 

Sin. 

Diff. 

INDEX. 


PAGE 

Aberration,  law  of  .  .160 
in  R.A.  and  Dec.  .  .  166,  292 
rigorous  reduction  for  .  .  307 
in  longitude  and  latitude  .  163 
constant  of  .  .  .165 
relation  to  sun's  parallax  .  166 
effect  of  motion  on  .  .  .  169 
of  the  planets .  .  .  .170 
diurnal 168 

Adjustment  of  discordant  quantities    50 

Airy,  work  of  ....  342 
star  catalogues  of  .  .  .  383 

Aldebaran,  declination  of   .         .         58 

Almanaque  Nautico,  mentioned.         14 

Almucantur,  defined  .         .  92 

Altitude,  defined  ...  94 
relation  to  time,  etc.  .  .  134 

Angle  of  the  vertical,  defined  .  145 
parallactic  (see  parallactic)  .  95 

Angles,  small,  used  for  their  sines 

or  tangents       ...  2 

development  of        ...  3 

Apparent  places  of  stars  .  .  289 
trigonometric  reduction  to  .  298 
reduction  for  nutation  .  289,  299 
for  aberration  .  .  292,  301 

for  parallax  ....  293 
graphic  methods  mentioned  .  316 
practical  methods  of  reduction  302 
second  order,  terms  of  .  .  306 
tables  for  computation  .  .  310 

Argelander,  Durchmusterung  of  351 
star  catalogue  .  .  .  382 

Astronomische  Gesellschaft,  fun- 
damental system  of  .         .       361 
catalogues  of  stars  .         .         ,       350 

Atmosphere,  density  of  .174 

laws  of  density         .         .         .183 

hypotheses  of  .         .185 

Newton's  hypothesis  of         .       185 

Bessel's  hypothesis  of  .         .       186 

Ivory's  hypothesis  of   .         .       187 

other  hypotheses  of      .         .       188 

equilibrium  of,  formulae  .       183 

adiabatic      .         .         .         .180 

pressure  of,  law  of  .         .         .       175 


Auwers,  investigates  systematic 

corrections        .         .         .       354 
day-numbers,  1726-1750.         .       316 
fundamental  system  of  the  A.G.    361 
of  Berliner  Jahrbuch   .         .       362 
form  of  corrections  to  R.A.     .       358 
method  of  reduction  for  pre- 
cession     ....       284 
star  catalogues  by  .         .        380,  381 
Azimuth,  defined         ...         94 
computation  of         .         .         .       131 
B.C.,  how  used  in  astronomy       .       123 
Bessel,  astronomical  constants  of      255 
Tabulae  Regiomontanae          254,  311 
founder  of  German  school        .       343 
meridian  circle  .      .         .  343 

Fundamenta  Astronomiae        .       380 
star  tables       .         .         .         .311 
Bond,  zones  observed  by     .         .      350 
Boss,  fundamental  system  of     361,  363 
Bradley,  work  of         ...       340 
Cape  of  Good  Hope,  Observatory  of  345 
star  catalogues  of    .         .        386,  387 
Catalogues  of  stars,  list  of          .       380 
mode  of  using  .         .         .       364 

systematic  corrections  to         .       367 
reductions  for  precession          .       371 
weights  of       ....       367 
methods  of  combining     .        364,  369 
Celestial  sphere,  conception  of   .         89 
Circle,  graduated,  use  of     .         .       331 
Clock  error,  method  of  determining  322 
Colure,  defined   ....         94 
Conditional  equations,  how  arising     63 
combination  of         ...         70 
Connaissance  des  Temps,  mentioned    14 
Constants,   table  of  (see  preces- 
sion, nutation,  etc.) .         .       393 
Coordinates,  defined   ...        87 
special  systems        ...         94 
geocentric,  on  earth's  surface  .       144 
equatorial  and  ecliptic    .         .         99 
transformation  of    .         .        100,  101 
spherical,  defined    ...         90 
Correction  of  provisional  elements      66 
systematic,  to  star  catalogues        367 


442 


INDEX 


Curvature  of  a  ray  of  refracted 

light          ....       198 

irregular          ...         .         .200 

Date,  mean          ....         82 

of  star  observations          .         .       373 

Day,  solar,  as  unit  of  time  .        114,  124 

sidereal,  defined      .         .         .115 

Day-numbers,  Besselian       .         .       294 

independent    ....       295 

Declination,  defined    ...         95 

trigonometric  reduction  of     268,  270 

determination  from  observation     331 

Density  of  the  atmosphere  .         .       174 

hypotheses  as  to  .         .       185 

relation  to  index  of  refraction        193 

Derivatives,    relation    to  speeds 

and  units ....  9 

Development,  numerical  .  .  36 
in  powers  of  time  ...  37 
Deviation,  local,  of  plumb  line  .  142 
Differences,  formation  and  use  of  17 
Differencing,  detection  of  errors 

by     ....         .         19 

Distribution  of  errors  in  magnitude    55 

Dip  of  the  horizon       .         .         .       201 

Durchmusterung,  described        .       351 

Earth,  figure  and  dimensions  of  .        141 

Ecliptic,  defined         ..       &        \         93 

motion  of         ....       229 

numerical  computation  of    .       231 

reference  to  fixed  plane  of        .       232 

obliquity  of,  defined        .  93 

numerical  values .         .         .       238 

Elements,  provisional,  correction  of    66 

Epoch,  mean        .         .         .         .         82 

of  observations        .         .         .       365 

Ephemerides,  astronomical,  list  of        14 

of  fixed  stars .          .         .        303,  310 

Equation  of  time         .         .         .116 

Equations  of  condition,  origin  of        63 

normal,  formation  of  .         70 

solution  of  normal  equations  .         73 

weights  of  unknowns       .         .         77 

Equator,  celestial,  defined  .         .         92 

obliquity  of,  to  fixed  ecliptic  .       244 

referred  to  ecliptic .         .         .       239 

positions  of,  at  various  epochs       238 

Equinox,  defined          .         .         93,  225 

absolute  reduction  of  R.A.'s  to     325 

motion  of,  defined  .         .         .       226 

Era,  Christian     .         .         .         .123 

Error,  probable,  defined      .         .         43 

of  a  product     ....         44 

of  a  sum  .....         44 

of  a  linear  function          .         .         45 

of  an  arithmetical  mean  .         .46 

of  a  weighted  mean          .         .         49 

determination  of     .         .     53,  57,  78 


Error  when  weights  are  unequal          60 

mean,  defined ....         54 

mean  and  probable  distinguished    60 

average,  defined      ...         54 

Errors,  of  function  and  variable  8 

small,  unavoidable  ...  6 

systematic,  defined          .         .         40 

fortuitous,  defined  .  .         41 

detection  of,  by  differencing   .          19 

law  of  distribution  of  .         56 

Finlay,  his  day-numbers  mentioned  316 

Flamsteed,  his  star-catalogues    .       339 

Fundamental  systems  of  star-places   361 

Gaussian  equations  in  Spherical 

Trigonometry   .         .         .104 

Geocentric  latitude      .         .         .143 

coordinates,  formulae  for         .        144 

Geographic  latitude    .         .         .144 

Geoid,  defined     .         .         .         .       141 

dimensions  and  form        .         .       146 

Gill,  work  at  Cape       .         .         .       347 

star-catalogues  by   .         .         .       387 

Graduated  circle,  use  of  .331 

Greenwich  observations,  history       339 

star-catalogues         .         .         .       383 

Groombrigde  1 830,  proper  motion  of   262 

reduction  of     .         .         .         .       272 

Hour-angle,  defined    ...         95 

relation  to  time        .         .         .117 

Horizon,  defined          .         .  92 

dip  and  distance  of .         .         .       201 

Hudson,  T.  C.,  Star  Facilitator  of     316 

Index  of  refraction  of  air    .         .       193 

Infinitesimals,  treatment  of  small 

quantities  as     .         .         .  1 

Interpolation,  simplest  form  of  .         15 

Hansen's  formulae  of       .         .         22 

Sterling's  formulae  of      .         .         25 

Bessel's  formulae  of  25 

to  halves  ....         26 

to  thirds  ....         29 

to  fourths        ....         33 

to  fifths   .         .         .     *    .         .         34 

Ivory's  hypothesis,  development  of  215 

Jahrbuch,  Berliner,  fundamental 

system  of .         .         .         .362 

Lacaille,  observations  of     .         .       345 

Latitude,  celestial,  defined  .  88,  95 

precession  in    .         .         .         .       285 

terrestrial,    astronomical    and 

geographic         .         .         .143 
Least  squares,  method  of    .         .         40 
books  upon      ....         84 
Leverrier,  work  at  Paris  obser- 
vatory      ....       350 
constant  of  precession      .         .       255 
star-tables        ....       315 
Light,  velocity  of        .         .         .165 


INDEX 


443 


PAGE 

Logarithms,  imperfection  of 

number  of  figures  required       .  7 

tables  of;  list  .         .         .          13 

3-place  tables  ....       435 

Longitude,  celestial,  defined  .  88,  95 
precession  in  .  .  .  .  285 
of  sun,  from  observations  .  326 
mean,  of  moon,  tables  of  .  403 
terrestrial,  relation  to  time  117,  119 

Longitude     and     Latitude    into 

R.A.  and  Dec.  .         .       101,  429 
effect  of  small  changes  in        108,  432 

Mean,  the  arithmetical  .  .  42 
by  weights  ....  46 

Mean  place  of  star,  defined  ,  259 
systematic  corrections  of  .  351 
fundamental  systems  of  .  .  360 

Meridian  circle,  ideal .  .  .  331 
measurement  with  .  .  .  333 
systematic  errors  of  .  .  336 

Meridian,  defined        ...         93 

Mirage,  how  produced        .         .       200 

Moon,  time  of  culmination .  .  129 
time  of  rising  and  setting  .  136 
semi-diameter  of  .  .  157 

parallax  of       .         .         .         .       155 
mean  longitude  of   .         .         .       403 

Nadir,  defined  .  ....  92 
practical  use  of  .  .  335 

Newcomb,  fundamental  system  .       362 

Node  of  moon,  table  of         .         .       403 

Normal  equations,  formation  and 

solution    ....         70 

Nutation,  described  .  .  .246 
constant  of,  defined  .  .  249 
theory  of  ....  249 
numerical  expression  of  .  251,  434 
in  longitude  and  obliquity  .  252,  253 
in  R.A.  and  Dec.  .  .  289,  299 
trigonometric  reduction  for  .  299 
in  sidereal  time  .  .  .  121 

Obliquity  of  the  ecliptic,  defined  93 
motion  of  ....  237 
numerical  values  of  .  .  238 

Oppolzer,  investigates,  nutation 

and  precession .         .         .       255 

Parallactic  angle,  defined  .  .  95 
computation  of  ...  131 

Parallax,  defined  .  .  .148 
horizontal,  defined  .  .  .  149 
annual,  formulae  for  .  .  293 
of  star,  formulae  .  .  .  293 
in  altitude  .  .  .  148,  150 
in  R.A.  and  Dec.  .  .  .151 
of  the  moon  .  .  .  .155 
of  sun  and  planets  .  .  .157 
solar,  relation  to  aberration  .  166 

Perigee  of  moon,  table  of    .         .       403 


Peters  investigates  nutation 
Planets,  figure  of  disc  of  . 

semi-diameter  of  . 

parallax  of  .  . 

Plumb-line,  deviation  of  . 
Polar  distance,  defined 


PAGE 

253 

158 
159 
157 
142 

89 


Polar    stars,     determination    of 

mean  places     .         .         .       376 

Polaris,  the  pole  star  ...  94 
nearest  approach  of  to  pole  94,  277 

Pole,  general  definition  .  .  91 
celestial,  defined  ...  93 
law  of  motion  of  ...  227 

Pond,  work  of  .  .  .  .341 
star  catalogues  of  .  .  .  382 

Position-angle,  defined  .  .  Ill 
formulae  for  .  .  .  .  112 
differential  of  .  .  .  .1  13 

Poulkova,  Observatory  of,  founded  344 
instruments  and  history  of  .  344 
star-catalogues  of  .  .  .  383 

Precession,  described  .  .  226,  227 
law  of  .....  227 
general,  expression  for  .  .  235 
luni-solar,  defined  -  .  .  226 
planetary,  defined  .  .  .  226 
numerical  value  of  .  236,  242 
expression  for  .  .  235,  242 
annual  and  cent.,  of  stars  .  278 
secular  variation  of  .  .  279 
in  longitude  and  latitude  .  285-87 
values  of  constants  of  .  255,  394 

Precessional  constant          .         .       228 
motions,  combined  effect         ".       233 
expressions  for     .         .         .       235 
numerical  values  of      .        236,  394 
table  of  .         .         .406 

secular  variations  of,  table  .       407 

Pressure-height  of  atmosphere    .       177 

Prime  vertical,  defined        .         .         93 

Proper  motions  of  stars,  theory  of  260 
rigorous  reduction  for  .  .  263 
secular  variations  of  .  .  262 
change  of,  by  precession  .  264 

trigonometric  reduction  of  .  271 
in  star-catalogues  .  .  .  365 
determination  from  observa- 

tions        .         •        .        365,  366 

Pulkowa  ;  see  Poulkova 

Rectangular  coordinates  .  .  97 
relations  to  polar  ...  98 
differentials  of  ...  98 

Refraction,  index  of  .  •'  .  193 
astronomical,  cause  of  .  .  173 
at  small  zenith  distances  .  190 
differential  equation  of  .  .  207 
integration,  process  of  .  .  209 
development  in  series  .  .  211 


444 


INDEX 


Refraction,    form   in    which  ex- 
pressed    .         .         .         .194 
practical  determination  of       .        196 
function  of  temp,  and  pressure 

196,  218 
general  formulae  for        .         .       203 

table  of 433 

tables  of,  construction  of  .  220 
curvature  of  ray  produced  by.  198 
terrestrial  .  .  .  .199 
works  on  .  *.  .  .  223 
Residuals,  in  least  squares .  .  59 
Right  Ascension,  defined  .  .  95 
determination  from  observation  320 
systematic  correction  to  .  .  357 
reference  to  equinox  .  .  325 
conversion  to  longitude  and 

latitude  ....  102 
Secular  variation,  of  precessions  279 
Semi-diameter  of  moon,  numerical 

value  of    .         .         .         .       157 

of  planets         .         .         ^^  •       1 58 

time  of  passage  of  .         .         .       139 

Sidereal  time,  defined         *        .       115 

relation  to  R.A.       ,         .         .       115 

conversion  of  .         .         .         .       121 

table  for .         .         .         .        398,  399 

Speed  of  variation       .         .         .  9 

Sphere,  celestial,  defined    .         .         89 

Spherical  coordinates,  theory  of .         87 

apparent,  defined    ...         87 

geocentric,  defined  .      -,.  87 

relation  to  rectangular     .         .         97 

differentials  of          ...         98 

Spherical  trigonometry,  infinitesimal  1 1 

Spherical   triangle,   differentials 

of  parts     .         .         .10,  396 
formulae  of  solution         .         .       394 
Squares,  Least    ....        40 
Star,  mean  place  of,  defined        .       259 
proper  motions,  general  theory      260 
reduction  for,  rigorous    .         .       263 
approximations  to  .         .         .       264 
annual  precession  of         .         .279 
secular  variation  of          .         .281 
third  term  of  .         .         .         .282 
See  also  Apparent  places,  Mean 
places,  Precession,  Proper 
motion 

Star-places,  fundamental  systems  of  361 

determination  of,  from  catalogues  371 

Star-positions,  theory  of     .         .       259 

centennial  motion  of        .         .       279 

determined  from  observations        317 


Stone,    E.    J.,    tables    for    star- 
constants  .         .         .       315 
Cape  catalogues  by          .         .       387 
Struve,  F.  W.,  founds  Poulkova 

Observatory      .         .         .344 
Struve,  Otto,  constant  of  preces- 
sion .         .  .        355,  394 
Sub  Polo  (S.  P. ),  defined     .         .         93 
Sun,  reference  of  stars  to    .         .       325 
Sunrise  and  sunset,  time  of        .       135 
Tables  of  logarithms,  list  of        .         13 
of  multiplication,  list  of  .         .         14 
Temperature  of  air,  relation  to 

pressure   .         .         .         .175 

Time,  general  measure  of    .         .       123 

chronological  reckoning  of       .       123 

mean,  defined          .         .         .116 

apparent,  defined    .         •.""     .       116 

sidereal,  defined      .         .         .       115 

of  mean  noon        .         .         .126 

relation  of  to  solar  time        .       114 

relation  to  longitude    .         .       117 

equation  of,  defined         .         .       116 

quantity    varying     uniformly 

with          ....         59 
problems  involving  .         .       126 

tables  of  conversion  of     .         397-402 

units  of 124 

determined  from  altitude         .       134 
Transit  instrument,  the  ideal     .       318 
Trigonometric  reduction  for  pre- 
cession     ....       265 
rigorous  formulae  for       .         .       267 
approximate  formulae  for         .       268 
tables  for          .         .         .         .419 
of  declination  ....       270 
Units,  use  of        ....  9 

Ursae  Minoris,  /3,  reduction  of   .       276 
Variation,  speed  of     .  .          9 

Vertical,  defined          .         .         .    * 
prime,  defined         .         .         .         93 
angle  of  .         .         .         .         .       145 
Weights,  relation  to  probable  error      47 
of  star  catalogues    .         .         .       368 
mean  by  .         .         .         .         .         46 
Year,  solar,  defined     .         .    ~  ' .       125 
time  of  beginning    .         .         .•       403 
length  of          ...        126,  393 
Years  B.C.,    astron.    and   chron. 

count  of  .  .  .  .  123 
Zenith,  defined  ....  92 
Zenith  distance  ....  95 
Zero,  the  absolute  .  .  .175 


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